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Как действительно понять нейронные сети и KAN на интуитивном уровне

Вот вы читаете очередную статью про KAN и ловите себя на мысли, что ничего не понимаете.

Знакомая ситуация?

Не переживайте, вы не одни. И дело тут не в вас, суть в том, что множество материалов описывают концепции по отдельности, не объединяя их в единую картину.

И чтобы решить эту проблему раз и навсегда, а также окончательно понять KAN, нам необходимо переосмыслив всё с нуля и постепенно двигаясь от базовых принципов линейной алгебры через нейронные сети. Завершив, обобщая всё с помощью множеств. В процессе мы также рассмотрим некоторые довольно уникальные и новые идеи!

Статья будет следовать данной структуре:

1. Основные термины и концепции

Прежде чем переходить к нейронным сетям, важно разобраться в ключевых концепциях линейной алгебры, в этом помогут видео с канала 3Blue1Brown Русский.

1.1 Вкратце о некоторых дополнительных концепциях

Векторные пространства и подпространства

Векторное (линейное) пространство – это множество, где можно складывать векторы и умножать их на числа, следуя определённым правилам. Например, множество всех векторов на плоскости, как векторов на графике, является векторным пространством.

Векторное (линейное) подпространство (или подмножество пространства) – это часть векторного пространства, которая сама является векторным пространством. Примером может служить линия в плоскости, проходящая через начало координат, или плоскость в трёхмерном пространстве, проходящая через начало координат.

Аффинные пространства и подпространства

По словам французского математика Марселя Берже, «Аффинное пространство – это не более чем векторное пространство, о начале координат которого мы пытаемся забыть, добавляя переносы (смещения) к линейным трансформациям».

$T(v)=\mathbf{A}\cdot \mathbf{v} + \mathbf{b} \\ где \ \ \mathbf{A}\cdot \mathbf{v} \ \ – \ это \ линейное \ преобразование$

Лебеговы пространства

Сейчас я приведу довольно упрощённое объяснение данных пространств.
Но главное – иметь хотя бы некоторое представление, так как они будут упоминаться в статье.

Ранее рассматривались конечномерные вещественные векторные пространства $\mathbb{R}^n,$ где для двух векторов $\mathbf{x}$ и $\mathbf{y}$ метрика Минковского задаётся следующим образом:

$\|\mathbf{x} - \mathbf{y}\|_p = \left( \sum_{i=1}^n |x_i - y_i|^p \right)^{\frac{1}{p}}$

В частных случаях p = 1 и p = 2 эта метрика соответствует манхэттенскому и евклидову расстояниям. Но нас интересует именно $1 \leq p < \infty.$

Норма вектора $\mathbf{x}$ определяется как расстояние Минковского до начала координат:

$\|\mathbf{x}\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{\frac{1}{p}}$

При этом любой вектор с конечным числом элементов можно рассматривать как дискретную функцию, возвращающую соответствующие элементы по индексу. Это понятие можно расширить на бесконечное множество элементов, задав норму функции на множестве

$\|f\|_p = \left( \int_S |f(x)|^p \, dx \right)^{1/p}$

Кстати, думаю, очевидно, что в дискретном случае шаг по равен 1.

В контексте L^p - пространств, если ограничить меру стандартной мерой Лебега (например, длина, площадь, объём) на $\mathbb{R}^n,$ то функция f(x) принадлежит L^p, если норма функции сходится к конечному числу:

$\|f\|_p = \left( \int_S |f(x)|^p \, dx \right)^{1/p}< \infty$

К тому же в общем случае должен использоваться интеграл Лебега, но для упрощения можно ограничиться наиболее распространённым определением интеграла в виде интеграла Римана, хотя это и накладывает дополнительные ограничения на множество функций.

Кроме того, можно определить расстояние между функциями:

$\|f - g\|_p = \left( \int_S |f(x) - g(x)|^p \, dx \right)^{1/p}$

И так же скалярное произведение, но только при p = 2, то есть в Гильбертовом пространстве:

$\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, dx$

Компактные множества

По сути, это множество, которое является замкнутым и ограниченным.

Ограничённое множество – это множество, для которого существует конечное число такое что расстояние между любой парой точек этого множества не превышает

Замкнутое множество – это множество, содержащее все свои предельные точки (границы).

Например, множество  (0, 1)^n ограничено, но не замкнуто, так как не включает границы, а значит не компактно. А вот множество  [0, 1]^n уже компактно, так как включает границы.

2. Объяснение многослойного персептрона (MLP)

Многослойный персептрон (MLP) – является разновидностью сетей прямого распространения, состоящих из полностью связанных нейронов с нелинейной функцией активации, организованных как минимум в три слоя (где первый – входной). Он примечателен своей способностью различать данные, которые не являются линейно разделимыми.

Но прежде чем перейти к объяснению на примере, желательно разобраться с некоторыми теоретическими деталями, а именно с теоремами универсальной аппроксимации. Они являются семейством теорем, описывающих различные условия и подходы, при которых нейронные сети способны аппроксимировать широкий класс функций. Часто при упоминании для удобства их сводят к одной «теореме универсальной аппроксимации», ссылаясь, например, на теорему Цыбенко. Но такое упрощение сбивает с толку, так как не учитывает все детали и различные подходы. В данной статье я хочу уделить этим теоремам больше внимания.

Если говорить о них более подробно, то теоремы универсальной аппроксимации являются по своей сути теоремами существования и предельными теоремами. Они утверждают, что существует такая последовательность преобразований $\Phi_1, \Phi_2, \ \dots\ ,$ и что при достаточном количестве нейронов мы можем аппроксимировать любую функциюиз определённого множества функций с заданной точностью в пределах $\epsilon.$

Однако эти теоремы не предоставляют способа фактически найти такую последовательность. Кроме того, они не гарантируют, что какой‑либо метод оптимизации, например, обратное распространение, сможет найти эту последовательность. Обратное распространение и другие методы могут либо найти сходящуюся последовательность, либо застрять в локальном минимуме. Также теоремы не дают гарантии, что нейронная сеть любого конечного размера, например, состоящая из 1000 нейронов, будет достаточной для аппроксимации функции с желаемой точностью.

Теперь рассмотрим некоторые из них:

Теорема Цыбенко (1989)

Цыбенко доказал, что нейронные сети с одним скрытым слоем и произвольным числом нейронов, использующие в качестве функции активации сигмоиду:

$\sigma(x) = \frac{1}{1 + e^{-x}}$

Могут аппроксимировать любые непрерывные функции на компактном множествеТо есть функции $f \in C(K).$

Есть также пример ограниченной ширины и глубины:

Майоров и Пинкус (1999)

Майоров и Пинкус доказали, что нейронные сети с двумя скрытыми слоями и ограниченным числом нейронов в каждом слое могут быть универсальными аппроксиматорами в пространстве непрерывных функций на компактном множестве, если используются сигмоидные функции активации.

Во многих статьях под упоминанием теоремы универсальной аппроксимации, хотя и говорится в основном о теореме Цыбенко, часто рассматривается семейство нейронных сетей, определённых теоремами Хорника и Лешно:

Теорема Хорника 1989 и Теорема Хорника 1991

Курт Хорник и его коллеги расширили результаты Цыбенко и показали, что MLP с произвольным числом скрытых слоев и произвольным числом нейронов в каждом слое могут аппроксимировать любую непрерывную функцию с любой точностью на компактном множестве, рассматривая так же расширение на пространства L^p для любого $p \in [1, \infty).$ Они также доказали, что это свойство не зависит от конкретных функции активации, если она является непрерывной, ограниченной, неконстантой и неполиномиальной.

Расширение Лешно и соавторов (1992):

Лешно и соавторы расширили результаты Хорника, ослабив требования к функции активации, позволяя ей быть кусочно-непрерывной и лишь локально ограниченной, а также не является полиномом почти всюду, что позволяет включить частоиспользуемые функции активации, такие как ReLU, Mish, Swish (SiLU), GELU и другие.

2.1 Иллюстрация на примере

Перейдём теперь к более практическому объяснению. Рассмотрим популярный датасет make_circles, который представляет собой векторы в двумерном пространстве. В этом пространстве каждый вектор данных можно задать линейно, по крайней мере, с помощью двух базисных векторов.

Логистическая регрессия в классическом варианте, моделирующая логарифмические шансы события (отношение выполнения события к его невыполнению) как линейную комбинацию признаков:

$\ln\left(\frac{p}{1 - p}\right) = \mathbf{w} \cdot \mathbf{x} + \mathbf{b}$ $\frac{p}{1 - p} = e^{\mathbf{w} \cdot \mathbf{x} + \mathbf{b}}.$ $p = \sigma(\mathbf{w} \cdot \mathbf{x} + \mathbf{b}) = \frac{1}{1 + e^{-(\mathbf{w} \cdot \mathbf{x} + \mathbf{b})}}.$

По сути, является нейросетью с одним входным слоем, нулём скрытых слоёв и одним выходным слоем (также можно рассматривать и линейную регрессию, и прочие модели). Если для кого-то ноль скрытых слоёв кажется контринтуитивным, то это можно рассматривать как применение единичной матрицы к входному вектору, собственно, что и делает входной вектор.

$\mathbf{I}\cdot \mathbf{x} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} = \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix}$

Поскольку логистическая регрессия осуществляет линейное разделение данных в исходном пространстве, а наш набор данных make_circles является нелинейно разделимым в нем, необходимо преобразовать данные так, чтобы модель могла эффективно их разделить. Для этого добавим к нашей логистической регрессии один скрытый слой с тремя нейронами и линейной функцией активации $f_{\text{activation}}(x) = x.$ Посмотрим, как изменятся данные перед их подачей в логистическую регрессию:

$Входные \ данные: \\[10pt] \mathbf{x} = \begin{bmatrix} x_1\\ x_2 \end{bmatrix} \\$ $\text{Веса скрытого слоя:} \\[10pt] \mathbf{W} = \begin{bmatrix} w_{1,1} & w_{1,2} \\ w_{2,1} & w_{2,2} \\ w_{3,1} & w_{3,2} \end{bmatrix} \\$ $Смещение: \\[10pt] \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \\$ $Тогда \ выход \ скрытого \ слоя: \\[10pt] T(\mathbf{x}) = f_\text{activation}(\mathbf{W} \cdot \mathbf{x} + \mathbf{b}) \\[5pt] где \ f_\text{activation}(x) = x$ $Что \ эквивалентно: \\[10pt] T(\mathbf{\mathbf{x}}) = x_1 \cdot \begin{bmatrix} w_{1,1} \\ w_{2,1} \\ w_{3,1} \end{bmatrix} + x_2 \cdot \begin{bmatrix} w_{1,2} \\ w_{2,2} \\ w_{3,2} \end{bmatrix} + \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$

И также посмотрим на график:

После прохождения через скрытый слой с тремя нейронами, данные сохраняют свою двумерную структуру. Хотя теперь данные выражаются в виде трех координат, их эффективное пространство по-прежнему остается двумерным. Каждый вектор можно задать линейно, как минимум, с помощью двух трехмерных базисных векторов $\hat{i}$ и $\hat{j}$ . Это означает, что данные находятся в двумерном аффинном подпространстве трехмерного пространства, образуя плоскость. Таким образом, мы фактически применили аффинное преобразование к исходным данным, и наш классификатор линейно разделяющий данные все еще не может их разделить.

Теперь добавим сигмоидную нелинейную функцию активации вместо линейной к результатам преобразования и посмотрим на полученный результата. То есть:

$Тогда \ выход \ скрытого \ слоя: \\[5pt] T(\mathbf{x}) = \sigma(\mathbf{W} \cdot \mathbf{x} + \mathbf{b}) \\[5pt] где \ \sigma(x) = \frac{1}{1 + e^{-x}}$

Бинго! Мы получили данные, которые стали нелинейными по отношению к исходному пространству признаков. Как видно из иллюстрации, они по-прежнему не являются линейно разделимыми. В данной ситуации на помощь приходит обучение нейросети, например, с использованием метода обратного распространения. Этот процесс регулирует матрицу преобразования и смещения, фактически перемещая и поворачивая базисные вектора. Таким образом, после тренировки нейросети и преобразования данных можно ожидать, что классификатор в виде логистической регрессии сможет построить разделяющую плоскость и линейно разделить данные.

Подытожим наш пример:
Изначально мы работаем с набором векторов, расположенных в двумерном векторном пространстве.
Применяя аффинное преобразование, мы переводим их в двумерное аффинное подпространство трехмерного пространства. Затем, используя нелинейную функцию активации, мы преобразуем данные нелинейно относительно входного пространства.
Далее, в процессе обучения с использованием метода обратного распространения ошибки, мы настраиваем базисные вектора двумерного подпространства таким образом, чтобы после применения функции активации данные становились линейно разделимыми на выходе скрытого слоя. Это делает их пригодными для классификации с помощью логистической регрессии на выходном слое.

Пример выше геометрически демонстрирует, почему линейные функции активации не подходят для работы с нелинейно разделимыми данными. Хотя в нашем примере мы использовали линейную функцию активации $f_{\text{activation}}(x) = x.$ ситуация при общем случае $f_{\text{activation}}(x) = w \cdot x + b$ останется аналогичной. Линейные функции активации не способны "изгибать" пространство входных данных при преобразовании, что необходимо для разделения нелинейных классов. Именно поэтому для работы с такими данными требуется использование нелинейных функций активации, которые позволяют моделям нейросетей эффективно преобразовывать пространство признаков, обеспечивая возможность линейного разделения в преобразованном пространстве.

Стоит отметить, что после применения нелинейной функции активации данные не всегда преобразуются в пространство, размерность которого совпадает с количеством нейронов (или размерностью каждого базисного вектора). В данном примере, после применения сигмоидной функции активации данные образуют трёхмерное подпространство в трёхмерном пространстве, то есть фактически становятся трёхмерными, но это является частным случаем.

При этом нам не обязательно увеличивать размерность базисных векторов на скрытом слое, хоть это и дает больше возможностей модели. Мы можем оставить их количество неизменным с последующим нелинейным преобразованием или даже уменьшить размерность, также используя нелинейные функции. Концепция преобразования пространства остаётся применимой и актуальной даже в случае, если нейросеть имеет несколько скрытых слоёв, то есть является глубокой, или для других архитектур нейронных сетей.

Рассмотрим подробнее выходной слой для нашего примера. Ранее я описывал его как логистическую регрессию для бинарной классификации. Но что такое логистическая регрессия, если не аффинное преобразование с последующим применением функции активации, которой, как правило, является сигмоида. Сам слой выглядит следующим образом:

$\sigma\left(\mathbf{w}^\top\mathbf{x} +b\right) = \frac{1}{1 + \exp\left(-\left(\mathbf{w}^\top\mathbf{x} + {b}\right)\right)} \\[30pt] \mathbf{w}^\top\mathbf{x} +{b}\ = \begin{bmatrix} w_{1} & w_{2} & \dots & w_{n} \end{bmatrix} \begin{bmatrix} x_{i_1} \\ x_{i_2} \\ \vdots \\ x_{i_n} \\ \end{bmatrix} + b$

Итак, сначала мы выполняем скалярное произведение двух векторов. Это операция, по сути, преобразует- мерный вектор в одномерное пространство. Важно отметить, что скалярное произведение – симметричная операция, то есть результат не зависит от порядка векторов: $f(\mathbf{x}, \mathbf{y}) = f(\mathbf{y}, \mathbf{x})$ . Таким образом, не имеет значения, какой из векторов рассматривать как матрицу преобразования, а какой как преобразующий вектор. Однако в контексте нашей задачи матрицей преобразования, логичнее, если будут являться веса. После скалярного произведения мы добавляем смешение производя аффинное преобразование-мерного вектора в одномерное аффинное подпространство.

После этого применяем преобразование с помощью сигмоидной функции активации, которая, по сути, изменяет это одномерное подпространство. В итоге, математически имеем тот же скрытый слой, только в контексте нашей цели, того как преобразует этот слой наши данные, а так же вместе с функцией потерь, мы его интерпритируем для удобства как выходной слой.

2.2 Аффинное преобразование и применение функции активации как функции преобразования и суперпозиции суммы функций

Суперпозиция функций, известная так же как сложная функция или «функция функции», имеет вид f(g(x)), в случае суперпозиции суммы функций будет иметь вид f(g_1(x)+g_2(x)+...g_n(x)) .

И в целом это довольно очевидно по идее, что аффинное преобразование и применение функции активации схожи, так как обе функции и выполняют преобразование данных с одного пространства в другое и единственная разница как. Но сейчас мы их представим в виде суперпозиции суммы функций и применения функциональных матриц к векторам, что позволит более явно увидеть сходство и пригодится как основной момент в пункте с KAN.

Рассмотрим на примере который мы выше использовали.

Аффинное преобразование:

При аффинном преобразовании элементы результатирующего вектора будут иметь следующий вид :

$F_{\text{affine transform}}(\mathbf{Х}) = \begin{bmatrix} w_{1,1} x_{1} + w_{1,2} x_{2} + b_{1} \\ w_{2,1} x_{1} + w_{2,2} x_{2} + b_{2} \\ w_{3,1} x_{1} + w_{3,2} x_{2} + b_{3} \end{bmatrix}$

При этом данное аффинное преобразование $F_{\text{affine transform}}$ можно подать как сумму функций одной переменной, только функции будут вида $\phi_{j,i}(x) = w_{j,i}x_i +b_{j,i}$ (смещение для одной строки можно распределить как угодно по функциям внутри неё), и преобразованный вектор будет выглядеть следующим обаразом:

$F_{\text{affine transform}}(\mathbf{x}) = \begin{bmatrix} \phi_{1,1}(x_1) + \phi_{1,2}(x_2) \\ \phi_{2,1}(x_1) + \phi_{2,2}(x_2) \\ \phi_{3,1}(x_1) + \phi_{3,2}(x_2) \end{bmatrix} = \begin{bmatrix} X_{\text{AffineT}, 1} \\ X_{\text{AffineT}, 2} \\ X_{\text{AffineT}, 3} \end{bmatrix}$

Функция активации:

Преобразование с помощью функции активации будут иметь следующий вид:

$F_{\text{activation}}(F_{\text{affine transform}}(\mathbf{x})) = \begin{bmatrix} \sigma(X_{\text{AffineT}, 1}) \\ \sigma(X_{\text{AffineT}, 2}) \\ \sigma(X_{\text{AffineT}, 3}) \end{bmatrix}$

Но при этом это эквивалентно ситуации, где функции на главной диагонали являются функцией активации $\phi_{j = i}(x) = \sigma(x),$ а остальные $\phi_{j \neq i}(x) = 0 \cdot X_{AffineT \ i}$ .

$F_{\text{activation}}\left(F_{\text{affine transform}}(\mathbf{x})\right) = \begin{bmatrix} \phi_{1,1}(X_{\text{AffineT}, 1}) + \phi_{1,2}(X_{\text{AffineT}, 2}) + \phi_{1,3}(X_{\text{AffineT}, 3}) \\ \phi_{2,1}(X_{\text{AffineT}, 1}) + \phi_{2,2}(X_{\text{AffineT}, 2}) + \phi_{2,3}(X_{\text{AffineT}, 3}) \\ \phi_{3,1}(X_{\text{AffineT}, 1}) + \phi_{3,2}(X_{\text{AffineT}, 2}) + \phi_{3,3}(X_{\text{AffineT}, 3}) \end{bmatrix}$ $F_{\text{activation}}(F_{\text{affine transform} }(\mathbf{x})) = \begin{bmatrix} \sigma(X_{\text{AffineT} \ 1}) + 0 \cdot X_{\text{AffineT} \ 2} + 0 \cdot X_{\text{AffineT} \ 3} \\ 0 \cdot X_{\text{AffineT} \ 1} + \sigma(X_{\text{AffineT} \ 2}) + 0 \cdot X_{\text{AffineT} \ 3} \\ 0 \cdot X_{\text{AffineT} \ 1}) + 0 \cdot X_{\text{AffineT} \ 2} + \sigma(X_{\text{AffineT} \ 3}) \end{bmatrix}$

В общем виде каждое аффинное преобразование и применение функции активации можно рассмотреть подобным образом (если мы уже применили функциональную матрицу к вектору):

$\mathbf{x}_{l+1} = \sum_{i_l=1}^{n_l} \phi_{l,i_{l+1},i_l}(x_{i_l}) \\[20pt] =\left[ \begin{array}{cccc} \phi_{l,1,1}(x_{1}) & + & \phi_{l,1,2}(x_{2}) & + \cdots + & \phi_{l,1,n_l}(x_{n_l}) \\ \phi_{l,2,1}(x_{1}) & + & \phi_{l,2,2}(x_{2}) & + \cdots + & \phi_{l,2,n_l}(x_{n_l}) \\ \vdots & & \vdots & \ddots & \vdots \\ \phi_{l,n_{l+1},1}(x_{1}) & + & \phi_{l,n_{l+1},2}(x_{2}) & + \cdots + & \phi_{l,n_{l+1},n_l}(x_{n_l}) \\ \end{array} \right], \\[10pt] \quad \\[10pt] где \\[10pt] \phi_{l,i_{l+1},i_l}(x_{i_l}) = w_{l,i_{l+1},i_l }\cdot f_{\text{activation, }l}(x_{i_l})+ b_{l,i_{l+1}, i_l}\\[10pt] l - \text{индекс слоя}$

А так как мы рассматриваем $\phi_{l, l+1, i}(x_l)$ как функции одной переменной, то можно объединить предыдущую активацию и следующее аффинное преобразование в одно преобразование подобного вида (как в формуле выше) и интерпретировать его как слой.
Потому что, в любом случае, итоговая функция будет функцией одной переменной из-за того, что функция активации применяется только по диагонали (т. е. j = i ), а значит:

$\phi_{\text{linear}, j, i} (f_{\text{activation}}(x_i)) = \phi_{j, i} (x_i).$

По сути, определяя каждую функцию как $\phi_{j, i} (x_i) = w_{j, i} \cdot f_{\text{activation}}(x_i) + b_{j, i} .$

Хотя это всё и может показаться сперва контринтуитивным, так как мы в основном воспринимаем предыдущее аффинное преобразование и последующую активацию как один из слоёв, но это просто вопрос интерпретации.

3. Объяснение RBF нейронной сети (RBFN) и SVM, связь с MLP

В этом пункте я хочу показать, в чем же причина не использовать полиномиальные функции активации, связать SVM и MLP, пройти от линейного ядра к полиномиальному и Гауссовому, а также рассмотреть RBF нейронную сеть как альтернативу MLP, что будет важно в пункте с KAN. Бонусом также мы расширим RBF нейронные сети на многослойный случай и докажем для них теорему универсальной аппроксимации.

3.1 SVM как вид нейронной сети прямого распространения

В предыдущем пункте мы рассматривали частный случай сети прямого распространения в виде MLP, а теперь рассмотрим аналоги MLP в виде некоторых вариантов SVM и RBF нейронных сетей. Их любят разделять, но почему бы не рассмотреть их в контексте сходства.

SVM, как и MLP в классическом виде, строит гиперплоскость для разделения классов в задачах классификации. Единственное отличие заключается в том, что SVM при построении гиперплоскости необходимо максимизировать расстояние между самой гиперплоскостью и границами классов. Для этого применяется процесс квадратичной оптимизации.

Теперь рассмотрим математический аспект. При предсказении классов мы имеем в SVM такую формулу:

$f(\mathbf{x})_{\text{SVM}} = \sum_{i=1}^{n} \alpha_i y_i K(\mathbf{x}_i, \mathbf{x}) + b, \\где\ K - функция \ ядра, \ \mathbf{x}_i - \text{тренировочные вектора} , \mathbf{x} - входной\ вектор, \\[10pt]a_i - множители\ Лагранжа$

При тренировке решается, как уже говорилось, задача квадратичной оптимизации, но нас интересует то, что для этого нужно вычислить матрицу Грама, используя ядро попарно между нашими тренировочными данными – $K(\mathbf{x}_i, \mathbf{x}_i)$ .

Теперь, по поводу самой функции ядра, теоретически обоснованным является использование функций, которые попадают под определение теоремы Мерсера (Mercer's theorem), то есть являются симметричными $K(\mathbf{x}, \mathbf{x}') = K(\mathbf{x}', \mathbf{x}),$ положительно полуопределенными (матрица Грама имеет неотрицательные собственные значения). Это позволяет им выполнять так называемый ядерный трюк (kernel trick).

Часто используемыми подобными ядрами являются:

Линейное ядро:
$K_{\text{Linear}}(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\top \cdot\mathbf{y}, \quad \mathbf{x}, \mathbf{y} \in \mathbb{R}^d.$
Полиномиальное ядро:
$K_{\text{Poly }n}(\mathbf{x}, \mathbf{y}) = (\mathbf{x}^\top \cdot \mathbf{y} + r)^n, \quad \mathbf{x}, \mathbf{y} \in \mathbb{R}^d ,\quad r \geq 0, \quad n \geq 1.$
Гауссово (RBF) ядро:
$K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = \exp\left(-\frac{\|\mathbf{x} - \mathbf{y}\|^2}{\lambda}\right), \quad \mathbf{x}, \mathbf{y} \in \mathbb{R}^d, \quad \lambda > 0.$

Ядерный трюк (kernel trick) — способ вычисления скалярного произведения в пространстве, сформированном функцией неявного преобразования. И выглядит это следующим образом:

$K(\mathbf{x}, \mathbf{x}') = \varphi(\mathbf{x})^\top \cdot \varphi(\mathbf{x}').$

Функция неявного преобразования $\varphi$ принимает на вход наше множество данных и отображает его в новое пространство.

В данном пункте мы рассмотрим именно использование линейного ядра, где $\varphi_{\text{linear}}(\mathbf{x}) = \mathbf{x}.$

$K_{\text{Linear}}(\mathbf{x}, \mathbf{x}') = \varphi_{\text{linear}}(\mathbf{x})^\top \cdot \varphi_{\text{linear}}(\mathbf{x}') = \mathbf{x}^\top \cdot \mathbf{x'}$

Линейное ядро также попадает под определение теоремы Мерсера и способно выполнять ядерный трюк. Оно просто преобразует исходное пространство в себя и производит скалярное произведение.

Предлагаю рассмотреть SVM как нейросеть с тремя слоями, где входным слоем будет наша функция $\varphi_{\text{linear}}(\mathbf{x}).$ Скрытым слоем будет просто линейное преобразование. В этом слое матрица весов будет построена с использованием предварительно преобразованных тренировочных векторов, используя функцию $\varphi_{\text{linear}}(\mathbf{x}).$ Выходным слоем будет аффинное преобразование, где веса состоят из множителей Лагранжа и значений целевой функции для каждого тренировочного вектора.

При этом данный вариант мы не можем рассматривать как вариант MLP из-за того, что функция активации, то есть ядро является линейным. По сути, данная модель производит линейное и последующее аффинное преобразование и является полностью линейной.

Рассмотрим набор из тренировочных векторов:

$\mathbf{x}_{\text{train}, 1},\, \mathbf{x}_{\text{train}, 2},\, \dots,\, \mathbf{x}_{\text{train}, n}$ где каждый вектор $\mathbf{x}_{\text{train}, j} \in \mathbb{R}^d, \quad j = 1,,2,,\dots,,n.$

Также рассмотрим набор из тестовых векторов:

$\mathbf{x}_{\text{test}, 1},\, \mathbf{x}_{\text{test}, 2},\, \dots,\, \mathbf{x}_{\text{test}, m}$ где каждый вектор $\mathbf{x}_{\text{test}, q} \in \mathbb{R}^d, \quad q= 1,\,2,\,\dots,\,m.$

Формулу нашей модели можно тогда подать как:

$f(\mathbf{x})_\text{Linear SVM} = \mathbf{W}_2(\mathbf{W}_1 \cdot \varphi_\text{linear}(\mathbf{x}))+b_\text{out}$

Для наглядности можно записать матрицы в развернутом виде:

$\mathbf{W}_1 = \begin{bmatrix} \varphi(x_{\text{train},1}) \\ \varphi(x_{\text{train},2}) \\ \vdots \\ \varphi(x_{\text{train},n}) \end{bmatrix} \\[50pt] = \begin{bmatrix} \varphi(x_{\text{train},1,1}) & \varphi(x_{\text{train},1,2}) & \dots & \varphi(x_{\text{train},1,d}) \\ \varphi(x_{\text{train},2,1}) & \varphi(x_{\text{train},2,2}) & \dots & \varphi(x_{\text{train},2,d}) \\ \vdots & \vdots & \ddots & \vdots \\ \varphi(x_{\text{train},n,1}) & \varphi(x_{\text{train},n,2}) & \dots & \varphi(x_{\text{train},n,d}) \end{bmatrix} \\[50pt] = \begin{bmatrix} x_{\text{train},1,1} & x_{\text{train},1,2} & \dots & x_{\text{train},1,d} \\ x_{\text{train},2,1} & x_{\text{train},2,2} & \dots & x_{\text{train},2,d} \\ \vdots & \vdots & \ddots & \vdots \\ x_{\text{train},n,1} & x_{\text{train},n,2} & \dots & x_{\text{train},n,d} \end{bmatrix}$ $\mathbf{W}_2 = \begin{bmatrix} a_1 y_1 \\ a_2 y_2 \\ \vdots \\ a_n y_n \end{bmatrix}\\[15pt]$

Можно также переписать и саму формулу модели:

$f_\text{Linear SVM}(\mathbf{x}) = \mathbf{W}_2 \left( \begin{bmatrix} x_{\text{train}, 1, 1} & x_{\text{train}, 1, 2} & \dots & x_{\text{train}, 1, d} \\ x_{\text{train}, 2, 1} & x_{\text{train}, 2, 2} & \dots & x_{\text{train}, 2, d} \\ \vdots & \vdots & \ddots & \vdots \\ x_{\text{train}, n, 1} & x_{\text{train}, n, 2} & \dots & x_{\text{train}, n, d} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{d} \end{bmatrix} \right) + b_\text{out}\\[15pt]$

В обучении модели, конечно же, из-за использования квадратичной оптимизации, как было обсуждено в начале этого пункта, мы сначала строим матрицу Грама из тренировочных векторов.

$Матрица \ Грама \\[10pt] \mathbf{G} = \mathbf{W}_1 \cdot \varphi(\mathbf{x})\\[20pt] \mathbf{x}_{\text{input}} = \begin{bmatrix} x_{\text{train},1} \\ x_{\text{train},2} \\ \vdots \\ x_{\text{train},n} \end{bmatrix}\\[45pt]\mathbf{G} = \begin{bmatrix} x_{\text{train},1}^\top x_{\text{train},1} & x_{\text{train},1}^\top x_{\text{train},2} & \dots & x_{\text{train},1}^\top x_{\text{train},n} \\ x_{\text{train},2}^\top x_{\text{train},1} & x_{\text{train},2}^\top x_{\text{train},2} & \dots & x_{\text{train},2}^\top x_{\text{train},n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{\text{train},n}^\top x_{\text{train},1} & x_{\text{train},n}^\top x_{\text{train},2} & \dots & x_{\text{train},n}^\top x_{\text{train},n} \end{bmatrix}\\[15pt]$

После этого обучаем множители Лагранжа в матрице $\mathbf{W}_2$ , и можем тестировать модель.

$f_{\text{Linear SVM}}(x_{\text{test}, j}) = \\[10pt]= \mathbf{W}_2 \left( \begin{bmatrix} x_{\text{train}, 1, 1} & x_{\text{train}, 1, 2} & \dots & x_{\text{train}, 1, d} \\ x_{\text{train}, 2, 1} & x_{\text{train}, 2, 2} & \dots & x_{\text{train}, 2, d} \\ \vdots & \vdots & \ddots & \vdots \\ x_{\text{train}, n, 1} & x_{\text{train}, n, 2} & \dots & x_{\text{train}, n, d} \end{bmatrix} \begin{bmatrix} x_{\text{test}, j, 1} \\ x_{\text{test}, j, 2} \\ \vdots \\ x_{\text{test}, j, d} \end{bmatrix} \right) + b_\text{out}\\[15pt]$

Все то, что мы рассмотрели выше, можно подать в виде схемы:

$\text{Входные данные} \quad \mathbf{x} \quad \xrightarrow{\varphi} \quad \varphi(\mathbf{x}) \quad \xrightarrow{\cdot \mathbf{w}} \quad \varphi(\mathbf{x}) \cdot \mathbf{w} \quad \xrightarrow{\text{выход модели}} \quad \hat{y} \quad (1) \\[10pt] где \ \mathbf{w} = \varphi(\mathbf{x}_i), \ \mathbf{x}_i - \text{тренировочные вектора}\\[15pt]$

В реальности, конечно, будет использоваться ядро для уменьшения вычислительных затрат, и все будет выглядеть следующим образом:

$\text{Входные данные} \quad \mathbf{x} \quad \xrightarrow{K(\mathbf{x}, \mathbf{x}_i)} \quad K(\mathbf{x}, \mathbf{x}_i) \quad \xrightarrow{\text{выход модели}} \quad \hat{y} \quad (2) \\[10pt] где \ \mathbf{x}_i - \text{тренировочные вектора}\\[15pt]$

Что соответствует формуле, которую мы рассматривали в самом начале этого пункта:

Ну, или в случае с линейным ядром, мы имеем формулу:

$f(\mathbf{x})_{\text{Linear SVM}} = \sum_{i=1}^{n} \alpha_i y_i (\mathbf{x}_i\cdot \mathbf{x}) + b$

Несмотря на то, что структура модели для первой схемы и для второй одинакова, это является лишь частным случаем при линейном ядре, что будет видно в следующих примерах с другими ядрами.

3.1.1 Полиномиальное ядро

Возьмем вторую схему:

$\text{Входные данные} \quad \mathbf{x} \quad \xrightarrow{K(\mathbf{x}, \mathbf{x}_i)} \quad K(\mathbf{x}, \mathbf{x}_i) \quad \xrightarrow{\text{выход модели}} \quad \hat{y} \quad (2) \\[10pt] где \\\mathbf{x}_i - \text{тренировочные вектора}$

Теперь рассмотрим использование полиномиального ядра, в общем виде оно задано как:

$K_{\text{Poly }n}(\mathbf{x},\mathbf{y}) = (\mathbf{x}^\top \mathbf{y} + r)^n, \ \ \mathbf{x}, \mathbf{y} \in \mathbb{R}^d, \quad r \geq 0, \quad n \geq 1$

Где– показатель степени полиномиального ядра.

Упростим для примера сделав n = 2 и r = 0 :

$K_{\text{Poly }2}(\mathbf{x},\mathbf{y})=(\mathbf{x}^\top \cdot \mathbf{y})^2$

Представим теперь, как трехслойный MLP, используя вторую схему, тогда формула будет выглядеть так:

$f(\mathbf{x})_{\text{Poly SVM}} = \sum_{i=1}^{n} \alpha_i y_i K_{\text{poly }2}(\mathbf{x}_i, \mathbf{x}) + b = \sum_{i=1}^{n} \alpha_i y_i (\mathbf{x}_i\cdot \mathbf{x})^2 + b$

Фактически, мы просто заменяем функцию активации в предыдущем примере с линейным ядром $f_{\text{activation}}(x) = x$ на квадратичную $f_{\text{activation}}(x) = x ^2$ на скрытом слое. Таким образом, в скрытом слое будет выполняться аффинное преобразование с последующей квадратичной функцией активации.

Однако, как я упоминал в предыдущем пункте про MLP, не все нелинейные функции активации изменяют размерность преобразованных данных, равную размерности базисных векторов. В общем случае для двумерных векторов при использовании квадратичной функции активации они будут образовывать максимум трёхмерное подпространство в произвольном- мерном пространстве.

И вот тут первая схема нам и пригодится:

Поэтому сейчас мы разберёмся с функцией $\varphi_{\text{poly }n},$ чтобы понять, почему Хорник и Лешно утверждали, что нейронные сети с именно неполиномиальными функциями активации обладают свойством универсальной аппроксимации, а также почему использование полиномиальных функций активации в нейронных сетях не является популярным решением.

Для двумерных данных с полиномиальным ядром второй степени по определению функция $\varphi_{\text{poly }2}$ выглядит подобным образом:

$\varphi_{\text{poly }2}(\mathbf{x} ) = \begin{bmatrix}x_1^2 \\x_2^2\\ \sqrt{2}x_1x_2\end{bmatrix}$

Вот доказательство:

$\varphi_{\text{poly }2}(\mathbf{x}_n)^{\top} \varphi_{\text{poly }2}(\mathbf{x}_m) = \begin{bmatrix} x_{n,1}^2 & x_{n,2}^2 & \sqrt{2} x_{n,1} x_{n,2} \end{bmatrix} \cdot \begin{bmatrix} x_{m,1}^2 \\ x_{m,2}^2 \\ \sqrt{2} x_{m,1} x_{m,2} \end{bmatrix}\\ = x_{n,1}^2 x_{m,1}^2 + x_{n,2}^2 x_{m,2}^2 + 2 x_{n,1} x_{n,2} x_{m,1} x_{m,2} = (x_n \cdot x_m)^2$

И иллюстрация этого преобразования:

То есть, как мы можем увидеть, применение полиномиального ядра второй степени к двумерным данным всегда отображает их нелинейно в трёхмерное пространство и после этого производит скалярное произведение.

Для трёхмерных векторов с полиномиальным ядром второй степени:

$\varphi_{\text{poly }2}(\mathbf{x}) = \begin{bmatrix} x_{1}^2 \\ x_{2}^2 \\ x_{3}^2 \\ \sqrt{2} x_{1} x_{2} \\ \sqrt{2} x_{1} x_{3} \\ \sqrt{2} x_{2} x_{3} \end{bmatrix}$

Для двумерных векторов с полиномиальным ядром третьей степени:

$\varphi_{\text{poly }3}(\mathbf{x}) = \begin{bmatrix} x_{1}^3 \\ x_{2}^3 \\ \sqrt{3} x_{1}^2 x_{2} \\ \sqrt{3} x_{1} x_{2}^2 \\ \sqrt{3} x_{1}^2 \\ \sqrt{3} x_{2}^2 \\ \sqrt{6} x_{1} x_{2} \end{bmatrix}$ Доказательство и анализ для общего случая.

Полиномиальное ядро степени $n \in \mathbb{N}_0$ (неотрицательные целые числа) с параметром сдвига r = 0 определяется как:

$K_{\text{Poly }n}(\mathbf{x}, \mathbf{y}) = (\mathbf{x}^\top \mathbf{y})^n, \quad\mathbf{x}, \mathbf{y} \in \mathbb{R}^d, \quad n \in \mathbb{N}_0, \quad n \geq 1$

Необходимо доказать, что существует такая функция $\varphi_{\text{poly }n}: \mathbb{R}^d \rightarrow \mathbb{R}^M ,$ что:

$K_{\text{Poly }n}(\mathbf{x}, \mathbf{y}) = \varphi_{\text{poly }n}(\mathbf{x})^\top \varphi_{\text{poly }n}(\mathbf{y})$

и при этом размерностьпревышает размерностьдля $d \geq 2$ и $n \geq 2.$

Для этого мы воспозуемся мультиномиальной теоремой, которая обобщает биномиальную и триномиальную теоремы на случай произвольного количества слагаемых. Она гласит, что для любых вещественных чисел $x_1, x_2, \dots, x_d$ и натурального числа

$(x_1 + x_2 + \dots + x_d)^n = \sum_{k_1 + k_2 + \dots + k_d = n} \frac{n!}{k_1! k_2! \dots k_d!} x_1^{k_1} x_2^{k_2} \dots x_d^{k_d}$

где $k_1, k_2, \dots, k_d \in \mathbb{N}_0,$ и сумма $k_1 + k_2 + \dots + k_d = n.$

Подставляя это разложение обратно в выражение для ядра, получаем:

$K_{\text{Poly }n}(\mathbf{x}, \mathbf{y}) = \sum_{k_1 + k_2 + \dots + k_d = n} \frac{n!}{k_1! \, k_2! \, \dots \, k_d!} \prod_{i=1}^d (x_i y_i)^{k_i}$

Исходя из данного преобразования мы можем выразить $\varphi_{\text{poly }n}(\mathbf{x})$ как:

$\varphi_{\text{poly }n}(\mathbf{x}) = \left[ \varphi_{\text{poly }j}(\mathbf{x}) \right]_{j=1}^M, \quad M \to \infty$

Где каждая компонента $\varphi_{\text{poly }j}(\mathbf{x})$ соответствует уникальному набору мультииндексов $\mathbf{k}^{(j)} \in \mathbb{N}_0^M$ и определяется как:

$\varphi_{\text{poly } j}(\mathbf{x}) = \frac{x_1^{k_1^{(j)}} x_2^{k_2^{(j)}} \dots x_d^{k_d^{(j)}}}{\sqrt{k_1^{(j)}! \, k_2^{(j)}! \, \dots \, k_d^{(j)}!}}$

Теперь проанализируем что происходит спри различныхи

Если $d \geq 2$ и $n \geq 2:$
$M = \binom{n + d - 1}{d - 1} = \frac{(n + d - 1)!}{n! \, (d - 1)!} >d$
При фиксированной размерностии увеличении степенирастёт полиномиально относительно
При фиксированной степении увеличении размерностирастёт экспоненциально относительно
Для
Полиномиальное ядро первой степени эквивалентно линейному ядру $K_{\text{Linear}}$ и функция $\varphi_{\text{poly }n}$ преобразует данные в исходное пространство признаков.
Для
Функция $\varphi_{\text{poly }n}$ отображает входные данные в одномерное пространство признаков. Приэквивалент $\varphi_{\text{linear }}.$ При $n\geq2$ производит нелинейное преобразование в другое одномерное пространство.

То есть, если мы возьмём формулу SVM для первой схемы:

$f(\mathbf{x})_\text{Poly SVM} = \mathbf{W}_2(\mathbf{W}_1 \cdot \varphi_{\text{poly }n}(\mathbf{x}) + \mathbf{b}_\text{in})+b_\text{out}$

Матрица $\mathbf{W}_1,$ которая состоит из набора тренировочных векторов, предварительно преобразованных с помощью функции $\varphi_{\text{poly }n}(x),$ будет иметь размер $N \times M,$ где– количество тренировочных векторов, а– количество элементов преобразованного тренировочного вектора с помощью функции $\varphi_{\text{poly }n}(x),$

Если взять с $\varphi_{\text{poly }2}(x),$ ядром для двумерных входных векторов, матрица будет $N \times 3,$ для $\varphi_{\text{poly }3}(x)$ – $N \times 7$ для $\varphi_{\text{poly }2}(x)$ но для трёхмерных входных – $N \times 6.$

Ситуация с MLP и полиномиальной функцией активации аналогична, просто вместо тренировочных векторов мы используем заранее определённые веса.

Это и есть причина, почему Хорник и Лешно в своих работах указывали, что для универсальной аппроксимации функция активации нейронной сети не должна быть полиномиальной, так как это ограничивает возможности модели в увеличении размерности признаков после линейного или аффинного преобразования.

Непопулярность полиномиальных функций активации на практике объясняется двумя взаимосвязанными причинами. Во-первых, такие функции ограничивают размерность преобразованного пространства как мы уже обсудили. Во-вторых, при использовании метода обратного распространения ошибки для оптимизации возникает численная нестабильность при расчете градиентов по параметрам. С увеличением степени полинома, хотя размерность пространства растет и предоставляет модели больше возможностей, высокие степени могут вызывать нестабильность в вычислении градиентов. Таким образом, решая любую одну из проблем, мы всегда сталкиваемся с другой.

3.1.2 Гауссово ядро

Но перед тем как начать обсуждение этого пункта, необходимо ввести определение радиальной базисной функции в контексте данной статьи.

Радиальная базисная функция (RBF) будет являться вещественнозначной функцией, которая принимает вектор и вычисляет эвклидово расстояние до другого фиксированного вектора, который часто называют центром. То есть $f(\mathbf{x}) = \|\mathbf{x} - \mathbf{c}\|.$

В общем виде ее, конечно, можно расширить на различные виды норм для векторных пространств, а также на функциональные и другие виды.

Теперь перейдём к рассмотрению RBF (в данном случае гауссового) ядра и его вычислению скалярного произведения в бесконечномерном пространстве.

Доказательство того, что Гауссово ядро вычисляет скалярное произведение в бесконечномерном пространстве, и нахождение функции неявного преобразования.

Исходное выражение Гауссового ядра при $\lambda = 2:$

$K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = \exp\left( -\frac{1}{2}||\mathbf{x} - \mathbf{y}||^2 \right)$

При этом эвклидово расстояние между $\mathbf{x}$ и $\mathbf{y}$ можно разложить как:

$||\mathbf{x} - \mathbf{y}||^2 = ||\mathbf{x}||^2 + ||\mathbf{y}||^2 - 2 \mathbf{x}^\top \cdot \mathbf{y}$ $K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = \exp\left( -\frac{||\mathbf{x}||^2 + ||\mathbf{y}||^2 - 2 \mathbf{x}^\top \cdot \mathbf{y}}{2} \right)$ $K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = \exp\left( -\frac{||\mathbf{x}||^2}{2} \right) \cdot \exp\left( -\frac{||\mathbf{y}||^2}{2} \right) \cdot \exp\left( \mathbf{x}^\top \cdot \mathbf{y} \right)$

Ряд Тейлора для экспоненты выглядит следующим образом:

$\exp(\mathbf{x}) = \sum_{n=0}^\infty \frac{\mathbf{x}^n}{n!}$

Подставляя $z = \mathbf{x} \cdot \mathbf{y},$ получаем:

$\exp(\mathbf{x}^\top \cdot \mathbf{y}) = \sum_{n=0}^\infty \frac{(\mathbf{x}^\top \cdot \mathbf{y})^n}{n!}$

Вспомним определение мултиномиальной теоремы для любых вещественных чисел $x_1, x_2, \dots, x_d$ и натурального числа

$(x_1 + x_2 + \dots + x_d)^n = \sum_{k_1 + k_2 + \dots + k_d = n} \frac{n!}{k_1! k_2! \dots k_d!} x_1^{k_1} x_2^{k_2} \dots x_d^{k_d}$

где $k_1, k_2, \dots, k_d \in \mathbb{N}_0,$ и сумма $k_1 + k_2 + \dots + k_d = n.$

В нашем случае:

$(\mathbf{x} \cdot \mathbf{y})^n = \left( \sum_{i=1}^d x_i y_i \right)^n$

Применяя мультиномиальную теорему, получаем:

$(\mathbf{x} \cdot \mathbf{y})^n = \sum_{k_1 + k_2 + \dots + k_d = n} \frac{n!}{k_1! k_2! \dots k_d!} \prod_{i=1}^d (x_i y_i)^{k_i}$

Подставляя разложение экспоненты и результат применения мультиномиальной теоремы в выражение для $K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}),$ получаем:

$K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = C\cdot \sum_{n=0}^{\infty} \frac{1}{n!} \left( \sum_{k_1 + k_2 + \dots + k_d = n} \frac{n!}{k_1! k_2! \dots k_d!} \prod_{i=1}^d (x_i y_i)^{k_i} \right)\\[20pt]где \\[10pt]C = \exp\left( -\frac{||\mathbf{x}||^2}{2} \right) \cdot \exp\left( -\frac{||\mathbf{y}||^2}{2} \right)$

Упрощая, получаем:

$K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = C \cdot\sum_{k_1, k_2, \dots, k_d = 0}^{\infty} \frac{(x_1 y_1)^{k_1} (x_2 y_2)^{k_2} \dots (x_d y_d)^{k_d}}{k_1! k_2! \dots k_d!}$

Мы знаем, что:

$K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = \varphi_{\text{gaussian}}(\mathbf{x})^\top \varphi_{\text{gaussian}}(\mathbf{y})$

Исходя из преобразований выше мы можем определить $\varphi_{\text{Gaussian}}(\mathbf{x})$ как:

$\varphi_{\text{Gaussian}}(\mathbf{x}) = \exp\left( -\frac{1}{2} ||\mathbf{x}||^2 \right) \cdot \left[ \varphi_{\text{gaussian }j}(\mathbf{x}) \right]_{j=1}^M, \quad M \to \infty$

Где каждая компонента $\varphi_{\text{Gaussian }j}(\mathbf{x})$ соответствует уникальному набору мультииндексов $\mathbf{k}^{(j)} \in \mathbb{N}_0^M$ и определяется как:

$\varphi_{\text{gaussian }j}(\mathbf{x}) = \frac{x_1^{k_1^{(j)}} x_2^{k_2^{(j)}} \dots x_d^{k_d^{(j)}}}{\sqrt{k_1^{(j)}! \, k_2^{(j)}! \, \dots \, k_d^{(j)}!}}$

Таким образом, исходя из того, что функция $\varphi_{\text{gaussian}}(\mathbf{x})$ возвращает вектор с бесконечным количество компонент, а значит что любой входной вектор фиксированной размерности будет отображаться в бесконечномерное пространство признаков. Из этого следует, что используя гауссово ядро $K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) ,$ и выполняя ядерный трюк, мы фактически вычисляем скалярное произведение нелинейно преобразованных входных векторов в это бесконечномерное пространство.

Возьмем Гауссово ядро:

$K_{\text{Gaussian}}(\mathbf{x}, \mathbf{y}) = \exp\left( -\frac{||\mathbf{x} - \mathbf{y}||^2}{\lambda} \right)$

Возьмем также знакомую нам первую схему и рассмотрим её для Гауссового ядра:

Функция $\varphi_\text{gaussian}(x)$ при $\lambda = 2$ определяется как:

$\varphi_{\text{gaussian}}(\mathbf{x}) = \exp\left( -\frac{1}{2} ||\mathbf{x}||^2 \right) \cdot \left[ \varphi_{\text{gaussian }j}(\mathbf{x}) \right]_{j=1}^\infty$

Где каждая компонента $\varphi_{\text{gaussian }j}(\mathbf{x})$ соответствует уникальному мультииндексу $\mathbf{k}^{(j)} \in \mathbb{N}^\text{M}_0$ и определяется как:

$\varphi_{\text{gaussian }j}(\mathbf{x}) = \frac{x_1^{k_1^{(j)}} x_2^{k_2^{(j)}} \dots x_d^{k_d^{(j)}}}{\sqrt{k_1^{(j)}! \, k_2^{(j)}! \, \dots \, k_d^{(j)}!}}$

Пример для двух двумерных векторов – $\mathbf{x}^\top= \begin{bmatrix}x_1, x_2\end{bmatrix}, \ \mathbf{y}^\top = \begin{bmatrix}y_1, y_2\end{bmatrix}$ и M = 2.

	$\mathbf{k}^{(j)} = [k_1^{(j)}, k_2^{(j)}]$	$\varphi_{\text{Gaussian }j}(\mathbf{x})$
1	(0, 0)	$\frac{x_1^0 x_2^0}{\sqrt{0! , 0!}} = 1$
			2	(1, 0)	$\frac{x_1^1 x_2^0}{\sqrt{1! \, 0!}} = x_1$
						3	(0, 1)	$\frac{x_1^0 x_2^1}{\sqrt{0! \, 1!}} = x_2$
4	(2, 0)	$\frac{x_1^2 x_2^0}{\sqrt{2! \, 0!}} = \frac{x_1^2}{\sqrt{2}}$
			5	(1, 1)	$\frac{x_1^1 x_2^1}{\sqrt{1! \, 1!}} = x_1 x_2$
						6	(0, 2)	$\frac{x_1^0 x_2^2}{\sqrt{0! \, 2!}} = \frac{x_2^2}{\sqrt{2}}$

Собирая всё вместе, $\varphi_\text{gaussian}(\mathbf{x})$ можно записать как:

$\varphi_\text{gaussian}(\mathbf{x}) = \exp\left( -\frac{1}{2} (x_1^2 + x_2^2) \right ) \left[ 1, x_1, x_2, \frac{x_1^2}{\sqrt{2}}, x_1 x_2, \frac{x_2^2}{\sqrt{2}} \right]$

Аналогично для $\varphi_\text{gaussian}(\mathbf{y})$ :

$\varphi_\text{gaussian}(\mathbf{y}) = \exp\left( -\frac{1}{2} (y_1^2 + y_2^2) \right ) \left[ 1, y_1, y_2, \frac{y_1^2}{\sqrt{2}}, y_1 y_2, \frac{y_2^2}{\sqrt{2}} \right]$

Теперь вычислим скалярное произведение между $\varphi_\text{gaussian}(\mathbf{x})$ и $\varphi_\text{gaussian}(\mathbf{y})$ :

$K_\text{Gaussian}(\mathbf{x}, \mathbf{y}) = \varphi_\text{gaussian}(\mathbf{x})^\top \cdot \varphi_\text{gaussian}(\mathbf{y}) \\ = \exp\left(-\frac{1}{2} (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2)\right) \cdot \left(1 + x_1 y_1 + x_2 y_2 + \frac{x_1^2 y_1^2}{2} + x_1 x_2 y_1 y_2 + \frac{x_2^2 y_2^2}{2} \right) \\ = \exp\left(-\frac{1}{2} (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2)\right) \cdot \left(1 + \mathbf{x}^\top \mathbf{y} + \frac{(\mathbf{x}^\top \mathbf{y})^2}{2!}\right) \\ = \exp\left(-\frac{1}{2} (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2)\right) \cdot \sum_{r=0}^{2} \frac{ (\mathbf{x}^\top \mathbf{y})^r }{r!} \\ \approx \exp\left(-\frac{1}{2} (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2)\right) \cdot \exp(\mathbf{x}^\top \mathbf{y})$

Теперь рассмотрим на примере SVM, что происходит с тренировочными и входными векторами, и что мы получим на выходе перед подачей в линейный классификатор. Для большей наглядности и уменьшения количества индексов я буду использовать в качестве тренировочных векторов двумерные векторы: $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ в качестве тестового вектора – $\mathbf{x}_\text{test}.$

Вектор $\mathbf{a}$ задан подобным образом:

$\mathbf{a} = \begin{bmatrix} a_{1} \\ a_{2} \end{bmatrix}$ $\varphi_\text{Gaussian}(\mathbf{a}) = \exp\left(-\frac{1}{2}(a_1^2 + a_2^2)\right) \cdot \left[1, \, a_1, \, a_2, \, \frac{a_1^2}{\sqrt{2}}, \, a_1 a_2, \, \frac{a_2^2}{\sqrt{2}}, \, \dots \right] \\=C_\text{a} \cdot \left[1, \, a_1, \, a_2, \, \frac{a_1^2}{\sqrt{2}}, \, a_1 a_2, \, \frac{a_2^2}{\sqrt{2}}, \, \dots \right]$

для $\mathbf{b}, \mathbf{c}, \mathbf{d}$ и $\mathbf{x}_\text{test}$ все аналогично.

Теперь построим матрицу преобразования, используя тренировочные векторы:

$\mathbf{W}_1 = \begin{bmatrix} C_{\text{a}} & C_{\text{a}} a_{\text{1}} & C_{\text{a}} a_{\text{2}} & C_{\text{a}} \frac{a_{\text{1}} a_{\text{2}}}{\sqrt{1}} & C_{\text{a}} \frac{a_{\text{1}}^2}{\sqrt{2}} & C_{\text{a}} \frac{a_{\text{2}}^2}{\sqrt{2}} & \cdots \\ C_{\text{b}} & C_{\text{b}} b_{\text{1}} & C_{\text{b}} b_{\text{2}} & C_{\text{b}} \frac{b_{\text{1}} b_{\text{2}}}{\sqrt{1}} & C_{\text{b}} \frac{b_{\text{1}}^2}{\sqrt{2}} & C_{\text{b}} \frac{b_{\text{2}}^2}{\sqrt{2}} & \cdots \\ C_{\text{c}} & C_{\text{c}} c_{\text{1}} & C_{\text{c}} c_{\text{2}} & C_{\text{c}} \frac{c_{\text{1}} c_{\text{2}}}{\sqrt{1}} & C_{\text{c}} \frac{c_{\text{1}}^2}{\sqrt{2}} & C_{\text{c}} \frac{c_{\text{2}}^2}{\sqrt{2}} & \cdots \\ C_{\text{d}} & C_{\text{d}} d_{\text{1}} & C_{\text{d}} d_{\text{2}} & C_{\text{d}} \frac{d_{\text{1}} d_{\text{2}}}{\sqrt{1}} & C_{\text{d}} \frac{d_{\text{1}}^2}{\sqrt{2}} & C_{\text{d}} \frac{d_{\text{2}}^2}{\sqrt{2}} & \cdots \\ \end{bmatrix}$ $\mathbf{W}_1 \times \varphi_\text{gaussian}(\mathbf{x}_\text{test}) = \begin{bmatrix} C_x C_a \left( 1 + a_1 x_1 + a_2 x_2 + \frac{a_1^2 x_1^2}{2} + a_1 a_2 x_1 x_2 + \frac{a_2^2 x_2^2}{2} + \cdots \right) \\ C_x C_b \left( 1 + b_1 x_1 + b_2 x_2 + \frac{b_1^2 x_1^2}{2} + b_1 b_2 x_1 x_2 + \frac{b_2^2 x_2^2}{2} + \cdots \right) \\ C_x C_c \left( 1 + c_1 x_1 + c_2 x_2 + \frac{c_1^2 x_1^2}{2} + c_1 c_2 x_1 x_2 + \frac{c_2^2 x_2^2}{2} + \cdots \right) \\ C_x C_d \left( 1 + d_1 x_1 + d_2 x_2 + \frac{d_1^2 x_1^2}{2} + d_1 d_2 x_1 x_2 + \frac{d_2^2 x_2^2}{2} + \cdots \right) \end{bmatrix}$

Преобразование с помощью функции $\varphi_\text{gaussian}$ Гауссового ядра даёт нам нелинейное отображение вектора в бесконечномерное пространство. Это позволяет нам построить матрицу преобразования размером $N \times M$ , где – количество векторов для обучения в случае SVM, либо количество нейронов в более общем случае, а так же $M \to \infty.$ Кроме того, итоговый ранг матрицы, как и размерность результатирующего вектора, будут зависеть именно от значения Именно в этом заключается главная особенность использования Гауссового ядра в SVM. К примеру, выше мы преобразовали вектор $\mathbf{x}$ из двумерного исходного пространства нелинейно в четырёхмерное пространство.

В случае с SVM в примере выше, мы преобразуем тренировочные вектора $\mathbf{a},\mathbf{b}, \mathbf{c}, \mathbf{d}$ при помощи $\varphi_\text{gaussian},$ затем через матрицу преобразования создаём элементы, которые были созданы с помощью тех же тренировочных векторов и функции $\varphi_\text{gaussian},$ по сути создавая матрицу Грама. Затем эта матрица будет использоваться в процессе квадратичной оптимизации для обучения множителей Лагранжа в выходном слое. После обучения мы можем тестировать модель на новых векторах, таких как $\mathbf{x},$ как мы и сделали выше.

Если мы возьмем вторую схему, где мы рассматриваем напрямую применение ядра:

Тогда формула SVM будет выглядеть следующим образом:

$f(\mathbf{x})_{\text{SVM RBF}} = \sum_{i=1}^{n} \alpha_i y_i K_{\text{Gaussian}}(\mathbf{x}_i, \mathbf{x}) + b = \sum_{i=1}^{n} \alpha_i y_i \exp\left( -\frac{||\mathbf{x} - \mathbf{x}_i||^2}{\lambda} \right) + b$

Но в отличие от линейного или полиномиального ядра, мы не можем представить данную нейронную сеть как MLP, поскольку вычисляем квадрат евклидова расстояния, а не скалярное произведение. Мы вернёмся к этому и рассмотрим более подробно в пункте про KAN. При этом SVM с RBF ядром является подмножеством RBF нейронной сети с определённым подходом к её построению и обучению. Поэтому предлагаю перейти к рассмотрению самой RBF нейронной сети.

3.2 RBF нейронная сеть

В предыдущем абзаце я уже говорил, что RBF-нейронная сеть применяет аналогичный подход как $f(\mathbf{x})_{\text{SVM RBF}},$ но более обобщённо: вместо тренировочных данных мы можем использовать любые случайные веса, поскольку они будут оптимизироваться в процессе обучения. Также нет необходимости фокусироваться на задаче максимизации расстояния между классами, использовании множителей Лагранжа или квадратичной оптимизации.

Таким образом, $f(\mathbf{x})_{\text{SVM RBF}}$ можно рассматривать как частный случай множества RBF-нейронных сетей. При этом любые RBF нейронные сети являются альтернативой MLP, так как если мы ее представим как композицию суммы функций, то каждая функция является не $w \cdot x$ , а ||w-x|| .

Одна из пионерских работ в области RBF нейронных сетей была написана Дж. Парком и В. Сандбергом. Там также предоставляется теорема универсальной аппроксимации для ограниченного количества слоев и произвольного количества нейронов.

Она гласит, что если ядро $K_\text{RBF}$ является интегрируемым, ограниченным и интеграл на всей области определения не нулевой, тогда:

$f_\text{Park, Sandberg}(\mathbf{x}) = \sum_{i=1}^{M} w_{i} \cdot K_\text{RBF}\left(\frac{\mathbf{x} - \mathbf{c}_{i}}{\lambda_i}\right)$

является универсальным аппроксиматором в пространствах $L^{p}(\mathbb{R}^{r})$ для любого $p \in [1, \infty)$ и пространтсвах непрерывных функций $C(\mathbb{R}^r).$

По сути, выполняя данное преобразование, данная нейронная сеть способна аппроксимировать широкий ряд функций:

$f_\text{Park, Sandberg}(\mathbf{x}) = \begin{bmatrix} w_1 & w_2 & \dots & w_M \end{bmatrix} \begin{bmatrix} K_\text{RBF}\left(\frac{\mathbf{x} - \mathbf{c}_1}{\lambda_1}\right) \\ K_\text{RBF}\left(\frac{\mathbf{x} - \mathbf{c}_2}{\lambda_2}\right) \\ \vdots \\ K_\text{RBF}\left(\frac{\mathbf{x} - \mathbf{c}_M}{\lambda_M}\right) \end{bmatrix}$

Где $c_{i}$ – это центры RBF ядра (веса), $\lambda_i$ – ширина ядра.

При этом, несмотря на частое использование Гауссового ядра, существуют и другие известные ядра, которые также могут выполнять скалярное произведение в бесконечномерном пространстве, а именно:

Inverse Quadratic RBF
$K_{\text{IQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{1 + (\lambda \cdot \mathbf{\|x} - \mathbf{y}\|)^2}$
Inverse Multiquadric RBF
$K_{\text{IMQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{1 + (\lambda\cdot \| \mathbf{x} - \mathbf{y} \|)^2}}$

И сейчас я приведу для каждого доказательство:

Inverse Quadratic RBF

$\| \mathbf{x} - \mathbf{y} \|^2 = \| \mathbf{x} \|^2 + \| \mathbf{y} \|^2 - 2 \mathbf{x}^\top \cdot \mathbf{y}$ $K_{\text{IQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{1 + \lambda^2 (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2 - 2 \mathbf{x}^\top \cdot \mathbf{y})} = \frac{1}{a - b (\mathbf{x}^\top \cdot \mathbf{y})}$

где и равны:

$a = 1 + \lambda^2 (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2), \quad b = 2\lambda^2$

Приведение к форме, удобной для разложения:

$K_{\text{IQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{a - b (\mathbf{x}^\top \cdot \mathbf{y})} = \frac{1}{a} \cdot \frac{1}{1 - \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y})}$

При условии, что:

$\left| \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y}) \right| < 1$

Можем использовать разложение в геометрическую прогрессию:

$\frac{1}{1 - z} = \sum_{j=0}^{\infty} z^j$

Таким образом, получаем:

$K_{\text{IQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{a} \sum_{j=0}^{\infty} \left( \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y}) \right)^j = \frac{1}{a} \sum_{j=0}^{\infty} \left( \frac{b}{a} \right)^j (\mathbf{x}^\top \cdot \mathbf{y})^j$

Определяем компоненты функции преобразования $\varphi_{\text{IQ}}(x)$ :

$\varphi_{\text{IQ}}(\mathbf{x}) = \left[ \varphi_{\text{IQ }j}(\mathbf{x}) \right]_{j=0}^{M}, \quad M\to\infty$

{}где:

$\varphi_{\text{IQ }j}(\mathbf{x}) = \sqrt{\frac{1}{a}} \left( \frac{b}{a} \right)^{j/2} \mathbf{x}^{\otimes j}$

Здесь:

$\mathbf{x}^{\otimes j}$ — тензорное произведение вектора $\mathbf{x}$ с самим собой раз.

Вычисление скалярного произведения:

$\varphi_{\text{IQ}}(\mathbf{x})^\top \cdot \varphi_{\text{IQ}}(\mathbf{y}) \\[20pt] = \sum_{j=0}^{M} \varphi_{\text{IQ }j}(\mathbf{x})^\top \cdot \varphi_{\text{IQ }j}(\mathbf{y}) = \frac{1}{a} \sum_{j=0}^{M} \left( \frac{b}{a} \right)^j (\mathbf{x}^\top \cdot \mathbf{y})^j = K_{\text{IQ}}(\mathbf{x}, \mathbf{y})\\[20pt]M \to \infty$

Inverse Multiquadric RBF

$\|\mathbf{x} - \mathbf{y}\|^2 = \|\mathbf{x}\|^2 + \|\mathbf{y}\|^2 - 2 \mathbf{x}^\top \cdot \mathbf{y}$

Тогда ядро можно записать как:

$K_{\text{IMQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{1 + \lambda^2 (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2 - 2 \mathbf{x}^\top \cdot \mathbf{y})}}$

Как и в доказательстве в предыдущем ядре:

$a = 1 + \lambda^2 (\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2), \quad b = 2 \lambda^2$

Тогда ядро переписывается как:

$K_{\text{IMQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{a - b (\mathbf{x}^\top \cdot \mathbf{y})}}$

Для удобства вынесем $\sqrt{a}$ за скобку:

$K_{\text{IMQ}}(\mathbf{x}, \mathbf{y}) = \frac{1}{\sqrt{a}} \left(1 - \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y})\right)^{-\frac{1}{2}}$

Теперь наша цель – разложить выражение в степенной ряд:

$\left( 1 - \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y}) \right)^{-\frac{1}{2}}$

Биномиальное разложение для $(1 - z)^{-j}$ при |z| < 1 и j = 0.5 :

$(1 - z)^{-1/2} = \sum_{j=0}^{\infty} \frac{(2j)!}{(j!)^2 4^j} z^j$

Исходя из разложения, компоненты $\varphi_{\text{IMQ}}(\mathbf{x})$ можно определить как:

$\varphi_{\text{IMQ}}(\mathbf{x}) = \left[\varphi_{\text{IMQ }j}\right]_{j=0}^M, \quad M\to\infty$

где:

$\varphi_{\text{IMQ }j}(\mathbf{x}) = \sqrt{\frac{(2j)!}{(j!)^2 4^j}} \left( \frac{b}{a} \right)^{j/2} \mathbf{x}^{\otimes j}$

Здесь:

$\mathbf{x}^{\otimes j}$ — тензорное произведение вектора $\mathbf{x}$ с самим собой раз.

Скалярное произведение функций неявного преобразования будет:

$\varphi_{\text{IMQ}}(\mathbf{x})^\top\cdot \varphi_{\text{IMQ}}(\mathbf{y}) \\[20pt] =\sum_{j=0}^{M} \varphi_{\text{IMQ }j}(\mathbf{x})^\top \cdot \varphi_{\text{IMQ }j}(\mathbf{y})= \sum_{j=0}^{M} \left( \frac{(2j)!}{(j!)^2 4^j} \left( \frac{b}{a} \right)^j (\mathbf{x}^\top \cdot \mathbf{y})^j \right) = K_{\text{IMQ}},\\[20pt]M \to \infty$

Проверка сходимости разложения

Несмотря на то, что видно, что данные ядра вычисляют скалярное произведение в бесконеченомерном пространстве, это происходит при условии, когда вышеперечисленные ряды сходятся, поэтому сейчас проверим условие сходимости.

Условие сходимости геометрической прогрессии для обоих ядер:

$\left| \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y}) \right| < 1$

По неравенству Коши-Буняковского:

$| \mathbf{x}^\top \cdot \mathbf{y} | \leq \| \mathbf{x} \| \| \mathbf{y} \|$ $\left| \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y}) \right| \leq \frac{b}{a} \| \mathbf{x} \| \| \mathbf{y} \| = \frac{2\lambda^2 \| \mathbf{x} \| \| \mathbf{y} \|}{1 + \lambda^2 (\| \mathbf{x} \|^2 + \| \mathbf{y} \|^2)}$

Наша задача — показать, что:

$\frac{2\lambda^2 \| \mathbf{x} \| \| \mathbf{y} \|}{1 + \lambda^2 (\| \mathbf{x} \|^2 + \| \mathbf{y} \|^2)} < 1$

Умножим обе части неравенства на знаменатель (так как знаменатель не равен нулю и всегда положителен):

$2\lambda^2 \| \mathbf{x} \| \| \mathbf{y} \| < 1 + \lambda^2 (\| \mathbf{x} \|^2 + \| \mathbf{y} \|^2)$ $\lambda^2 ( 2\| \mathbf{x} \| \| \mathbf{y} \| - \| \mathbf{x} \|^2 - \| \mathbf{y} \|^2)< 1$

Заметим, что:

$2\| \mathbf{x} \| \| \mathbf{y} \| - \| \mathbf{x} \|^2 - \| \mathbf{y} \|^2 = -(\| \mathbf{x} \| - \| \mathbf{y} \|)^2 \leq 0 < 1$

Так как неравенство является справедливым:

$\lambda^2 ( 2\| \mathbf{x} \| \| \mathbf{y} \| - \| \mathbf{x} \|^2 - \| \mathbf{y} \|^2)< 1$

То выполняется и данное условие:

$\left| \frac{b}{a} (\mathbf{x}^\top \cdot \mathbf{y}) \right| < 1$

3.2.1 Многослойная RBF нейронная сеть

То, что мы рассмотрели RBF-нейронную сеть с одним скрытым слоем, в целом можно экстраполировать на многослойную, где мы рекурсивно применяем набор RBF-ядер.

Для первого слоя:

$\mathbf{h}^{(1)} = \begin{bmatrix} K_\text{RBF}\left(\frac{\mathbf{x} - \mathbf{c}_1^{(1)}}{\lambda_1^{(1)}}\right) \\ K_\text{RBF}\left(\frac{\mathbf{x} - \mathbf{c}_2^{(1)}}{\lambda_2^{(1)}}\right) \\ \vdots \\ K_\text{RBF}\left(\frac{\mathbf{x} - \mathbf{c}_{M_1}^{(1)}}{\lambda_{M_1}^{(1)}}\right) \end{bmatrix}$

Для второго слоя:

$\mathbf{h}^{(2)} = \begin{bmatrix} K_\text{RBF}\left(\frac{\mathbf{h}^{(1)} - \mathbf{c}_1^{(2)}}{\lambda_1^{(2)}}\right) \\ K_\text{RBF}\left(\frac{\mathbf{h}^{(1)} - \mathbf{c}_2^{(2)}}{\lambda_2^{(2)}}\right) \\ \vdots \\ K_\text{RBF}\left(\frac{\mathbf{h}^{(1)} - \mathbf{c}_{M_2}^{(2)}}{\lambda_{M_2}^{(2)}}\right) \end{bmatrix}$

И так далее, до-го слоя:

$f_\text{MRBFN}(\mathbf{x}) = \begin{bmatrix} w_1 & w_2 & \dots & w_{M_L} \end{bmatrix} \cdot \mathbf{h}^{(L)}$

Здесь:

$\mathbf{c}_i^{(l}$ – центры RBF-нейронов на -м слое.
$\lambda^{(l)}_i$ – ширина ядра на -м слое.
– количество нейронов на -м слое.

Но увы, многослойные RBF-нейронные сети не получили широкой популярности в сообществе. Исследований на эту тему крайне мало, обсуждений на форумах тоже практически не встретишь. Более того, мне так и не удалось найти формального доказательства того, что многослойная RBF-сеть обладает свойством универсальной аппроксимации. Поэтому я приведу это доказательство сам.

Теорема универсальной аппроксимации многослойной RBF нейронной сети

В данном доказательстве за основу будет использоватся работа Дж. Парка и В. Сандберга – «Universal Approximation Using Radial-Basis-Function Networks»

Теорема 1: Плотность S_K в $L^p(\mathbb{R}^r)$ (Park, J.; I. W. Sandberg, 1991)

Пусть $K: \mathbb{R}^r \rightarrow \mathbb{R}$ – интегрируемая, ограниченная функция, непрерывная почти всюду и такая, что:

$\int_{\mathbb{R}^n} K(x) \, dx \neq 0$

Тогда семейство S_K, состоящее из конечных линейных комбинаций сдвигов функции

$S_K = \left\{ \sum_{i=1}^M w_i K(\mathbf{x} - \mathbf{c}_i) : M \in \mathbb{N},\ w_i \in \mathbb{R},\ \mathbf{c}_i \in \mathbb{R}^r \right\}$

плотно в $L^p(\mathbb{R}^r)$ для любого $p \in [1, \infty).$

Tеорема 2: Плотность в $C(\mathbb{R}^r)$ с метрикой (Park, J.; I. W. Sandberg, 1991)

Пусть $K: \mathbb{R}^r \rightarrow \mathbb{R}$ – интегрируемая, ограниченная и непрерывная функция, такая, что:

$\int_{\mathbb{R}^r} K(x) \, dx \neq 0$

Тогда семейство S_K, состоящее из функций вида:

$q(x) = \sum_{i=1}^M w_i K(x - c_i)$

где $M \in \mathbb{N},$ $w_i \in \mathbb{R},$ и $c_i \in \mathbb{R}^r,$ плотно в $C(\mathbb{R}^r)$ относительно метрики:

$d(f, g) = \sum_{n=1}^\infty 2^{-n} \frac{\| (f - g) \cdot 1_{[-n, n]^r} \|_\infty}{1 + \| (f - g) \cdot 1_{[-n, n]^r} \|_\infty}$

Теорема 3: Универсальная аппроксимация многослойными RBF сетями

Пусть $K: \mathbb{R}^r \rightarrow \mathbb{R}$ – интегрируемая, ограниченная и непрерывная функция, такая что:

$\int_{\mathbb{R}^r} K(x) \, dx \neq 0$

ине является константой. Предположим, что каждый скрытый слой сети имеет параметр сглаживания $\lambda_l > 0.$ Тогда семейство многослойных RBF сетей с произвольным числом слоев $L \geq 2$ и произвольным числом нейронов в каждом слое M_l плотно в $C(\mathbb{R}^r)$ относительно равномерной аппроксимации на компактных множествах.

Доказательство:

Докажем это методом математической индукции по числу слоев.

Базовый случай ( L = 2 ):

При L = 2 сеть представляет собой стандартную RBF сеть с одним скрытым слоем.
Согласно теореме 2, семейство S_K плотно в $C(\mathbb{R}^r).$
Следовательно, утверждение верно для L = 2.

Индуктивный шаг:

Предположим, что теорема верна для некоторого $L = k \geq 2,$ то есть любую функцию $f \in C(\mathbb{R}^r)$ можно аппроксимировать равномерно на компактных множествах с помощью сети с слоями.

Теперь докажем, что утверждение верно для L = k + 1.

Лемма 3.1: Композиция универсальных аппроксиматоров

Если функцию $f: \mathbb{R}^r \rightarrow \mathbb{R}$ можно аппроксимировать с произвольной точностью функциями из семейства а функцию $h: \mathbb{R} \rightarrow \mathbb{R}$ можно аппроксимировать с произвольной точностью функциями из семейства то композиция $h \circ f$ может быть аппроксимирована композициями функций из и функций из

Доказательство:

Пусть $\hat{f} \in A$ и $\hat{h} \in B$ такие, что

$\| f - \hat{f} \|_\infty < \delta, \quad \| h - \hat{h} \|_\infty < \delta$ $\| h \circ f - \hat{h} \circ \hat{f} \|_\infty \leq \| h \circ f - h \circ \hat{f} \|_\infty + \| h \circ \hat{f} - \hat{h} \circ \hat{f} \|_\infty$

Поскольку равномерно непрерывна на значениях функции существует L_h > 0 такое, что

$\| h \circ f - h \circ \hat{f} \|_\infty \leq L_h \| f - \hat{f} \|_\infty < L_h \delta$ $\| h \circ \hat{f} - \hat{h} \circ \hat{f} \|_\infty = \| h - \hat{h} \|_\infty < \delta$ $\| h \circ f - \hat{h} \circ \hat{f} \|_\infty < (L_h + 1) \delta$

Выбирая $\delta$ достаточно малым, мы можем сделать ошибку аппроксимации произвольно малой.

Лемма 3.2: Композиция в многослойных сетях

Если функция может быть аппроксимирована с точностью $\delta$ сетью с слоями, и функция идентичности $\text{id}: \mathbb{R} \rightarrow \mathbb{R}$ может быть аппроксимирована на компактном множестве с точностью $\delta$ дополнительным слоем (с использованием функций из S_K ), то сеть с k + 1 слоями аппроксимирует с точностью $(L_h + 1) \delta.$

Доказательство:

Пусть f_k – аппроксимация функции сетью сслоями, так что $| f - f_k |_\infty < \delta.$

Пусть– функция идентичности, а – её аппроксимация дополнительным слоем, так что $\| h - h' \|_\infty < \delta$ на значениях функции f_k.

Тогда $h' \circ f_k$ аппроксимируетс ошибкой $(L_h + 1) \delta,$ используя результат из леммы 3.1.

Лемма 3.3: Линейная комбинация универсальных аппроксиматоров

Линейная комбинация функций, каждая из которых может аппроксимировать любую функцию из определенного класса с заданной точностью, также может аппроксимировать любую функцию из этого класса с нужной точностью.

Доказательство:

Пусть f_i аппроксимирует с $\| f - f_i \|_\infty < \epsilon_i$ для $i = 1, \dots, N.$

Рассмотрим линейную комбинацию:

$F = \sum_{i=1}^N w_i f_i$

Выбирая коэффициенты c_i соответствующим образом и минимизируя $\epsilon_i,$ можно обеспечить $\| f - F \|_\infty < \epsilon$ для любого желаемого $\epsilon > 0.$

Завершение индуктивного шага:

По индукционному предположению существует f_k такая, что

$\| f - f_k \|_\infty < \frac{\epsilon}{2(L_h + 1)}$

Поскольку удовлетворяет условиям теоремы 2, функцию идентичности можно аппроксимировать на значениях функции f_k с точностью

$\delta = \frac{\epsilon}{2(L_h + 1)}$

Функция $f_{k+1} = h' \circ f_k$ аппроксимирует с точностью

$\| f - f_{k+1} \|_\infty \leq (L_h + 1) \delta = \epsilon / 2$

Используя лемму 3.3, можно настроить выходной слой (линейная комбинация выходов) для уменьшения ошибки до значения менее $\epsilon.$

Следовательно, сеть с k + 1 слоями может аппроксимировать с точностью $\epsilon.$

По индукции, теорема 3 верна для всех $L \geq 2.$

Следствие 3.1: Универсальная аппроксимация векторнозначных функций

Многослойные RBF сети с любым числом слоев $L \geq 2$ и любым числом выходов являются универсальными аппроксиматорами для векторнозначных функций $f: \mathbb{R}^r \rightarrow \mathbb{R}^m.$

Так как каждая компонента f_j векторнозначной функции может аппроксимироваться независимо многослойной RBF сетью, как показано в теореме 3. Объединяя аппроксимации всех компонент, получаем аппроксимацию функции с нужной точностью.

Следствие 3.2: Универсальная аппроксимация в пространствах

Многослойные RBF сети являются универсальными аппроксиматорами в пространстве $L^p(\mathbb{R}^r)$ для любого $p \in [1, \infty).$

Используя теорему 1 и теорему 3, мы видим, что семейство S_K плотно в $L^p(\mathbb{R}^r).$ Следовательно, многослойные RBF сети могут аппроксимировать любую функцию в $L^p(\mathbb{R}^r)$ с произвольной точностью.

Основная причина, почему этот подход остался в тени, кроется в том, как RBF-сети традиционно обучали. Обычно процесс выглядел так: данные сначала кластеризовали, центры кластеров брали в качестве параметров для RBF-нейронов, а затем настраивали веса только для линейного выходного слоя. В результате необходимость в многослойной архитектуре просто отпадала, делая её использование нелогичным и неоправданным.

Однако с развитием машинного обучения, когда стали массово применять метод обратного распространения ошибки, многослойные сети набрали большую популярность. Казалось бы, это должно было дать новый импульс и для многослойных RBF-сетей. Но увы, ассоциации с этим методом остались на уровне устаревшего подхода с одним скрытым слоем. Это, мягко говоря, несправедливо, ведь потенциал таких сетей гораздо шире.

В следующем разделе про KAN я более подробно покажу это, сравнив многослойные RBF-сети с другим, но подобным по концепции преобразования, вариантом KAN.

Реализация многослойной RBF нейронной сети на PyTorch.

Так как данная сеть может на практике сталкиватся с проблемой затухающих градиентов, я приведу два варианиа кода: с реализацией SkipConnection и DenseNet.

SkipConection MRBFN

import torch
import torch.nn as nn

# ---------------------
# Классический RBF слой
# ---------------------

class RBF(nn.Module):
    def __init__(self, in_features, out_features, basis_func, per_dimension_sigma=False):
        """
        Arg:
            per_dimension_sigma (bool): 
                Если True, для каждого выходного нейрона и каждого входного признака обучается отдельное значение σ.
                Если False, для каждого выходного нейрона используется одно общее значение σ для всех входных признаков.
        """
        super(RBF, self).__init__()
        self.in_features = in_features
        self.out_features = out_features
        self.per_dimension_sigma = per_dimension_sigma
        self.basis_func = basis_func

        if self.per_dimension_sigma:
            self.log_sigmas = nn.Parameter(torch.Tensor(out_features, in_features))

        else:
            self.log_sigmas = nn.Parameter(torch.Tensor(out_features))

        self.centres = nn.Parameter(torch.Tensor(out_features, in_features))
        self.reset_parameters()

    def reset_parameters(self):

        nn.init.normal_(self.centres, mean=0.0, std=1.0)
        nn.init.constant_(self.log_sigmas, 0.0)

    def forward(self, input):

        B = input.size(0)
        x = input.unsqueeze(1).expand(B, self.out_features, self.in_features)
        c = self.centres.unsqueeze(0).expand(B, self.out_features, self.in_features)

        if self.per_dimension_sigma:
            sigma = torch.exp(self.log_sigmas).unsqueeze(0).expand(B, self.out_features, self.in_features)
            distances = ((x - c).pow(2) / sigma).sum(dim=-1)

        else:
            sigma = torch.exp(self.log_sigmas).unsqueeze(0).expand(B, self.out_features)
            distances = (x - c).pow(2).sum(dim=-1) / sigma

        return self.basis_func(distances)

def gaussian_rbf(distances):
    return torch.exp(-distances)

      
# -----------------------------------------------------
# RBF слой с использованием идеи SkipConnection(ResNet)
# -----------------------------------------------------
  
class RBFResBlock(nn.Module):
    def __init__(self, in_features, out_features, basis_func, per_dimension_sigma=False):
        super(RBFResBlock, self).__init__()

        self.rbf = RBF(in_features, out_features, basis_func, per_dimension_sigma)

        if in_features != out_features:
            self.skip_connection = nn.Linear(in_features, out_features)

        else:
            self.skip_connection = nn.Identity()

    def forward(self, x):

        rbf_output = self.rbf(x)
        skip_connection_output = self.skip_connection(x)
        return rbf_output + skip_connection_output

# ------------------------
# Пример модели с 4 слоями
# ------------------------
      
class ResMRBFN(nn.Module):
    def __init__(self, in_features, hidden_units1, hidden_units2, hidden_units3, out_features, per_dimension_sigma=False):
        super(ResMRBFN, self).__init__()

        self.rbf_block1 = RBFResBlock(in_features, hidden_units1, gaussian_rbf, per_dimension_sigma)

        self.rbf_block2 = RBFResBlock(hidden_units1, hidden_units2, gaussian_rbf, per_dimension_sigma)

        self.rbf_block3 = RBFResBlock(hidden_units2, hidden_units3, gaussian_rbf, per_dimension_sigma)

        self.linear_final = nn.Linear(hidden_units3, out_features)

    def forward(self, x):

        x1 = self.rbf_block1(x)

        x2 = self.rbf_block2(x1)

        x3 = self.rbf_block3(x2)

        return self.linear_final(x3)

Dense MRBFN

import torch
import torch.nn as nn

# ---------------------
# Классический RBF слой
# ---------------------

class RBF(nn.Module):
    def __init__(self, in_features, out_features, basis_func, per_dimension_sigma=False):
        """
        Arg:
            per_dimension_sigma (bool): 
                Если True, для каждого выходного нейрона и каждого входного признака обучается отдельное значение σ.
                Если False, для каждого выходного нейрона используется одно общее значение σ для всех входных признаков.
        """
        super(RBF, self).__init__()
        self.in_features = in_features
        self.out_features = out_features
        self.per_dimension_sigma = per_dimension_sigma
        self.basis_func = basis_func

        if self.per_dimension_sigma:
            self.log_sigmas = nn.Parameter(torch.Tensor(out_features, in_features))

        else:
            self.log_sigmas = nn.Parameter(torch.Tensor(out_features))

        self.centres = nn.Parameter(torch.Tensor(out_features, in_features))
        self.reset_parameters()

    def reset_parameters(self):

        nn.init.normal_(self.centres, mean=0.0, std=1.0)
        nn.init.constant_(self.log_sigmas, 0.0)

    def forward(self, input):

        B = input.size(0)
        x = input.unsqueeze(1).expand(B, self.out_features, self.in_features)
        c = self.centres.unsqueeze(0).expand(B, self.out_features, self.in_features)

        if self.per_dimension_sigma:
            sigma = torch.exp(self.log_sigmas).unsqueeze(0).expand(B, self.out_features, self.in_features)
            distances = ((x - c).pow(2) / sigma).sum(dim=-1)

        else:
            sigma = torch.exp(self.log_sigmas).unsqueeze(0).expand(B, self.out_features)
            distances = (x - c).pow(2).sum(dim=-1) / sigma

        return self.basis_func(distances)

def gaussian_rbf(distances):
    return torch.exp(-distances)

      
# ---------------------------------------
# RBF слой с использованием идей DenseNet
# ---------------------------------------
     
class RBFDenseBlock(nn.Module):
    def __init__(self, in_features, out_features, basis_func, per_dimension_sigma=False):
        super(RBFDenseBlock, self).__init__()

        self.rbf = RBF(in_features, out_features, basis_func, per_dimension_sigma)

    def forward(self, x, previous_outputs):

        combined_inputs = torch.cat(previous_outputs, dim=1)

        rbf_densenet_output = self.rbf(combined_inputs)

        return rbf_densenet_output

      
class TransitionLayer(nn.Module):
    def __init__(self, in_features, out_features):
        super(TransitionLayer, self).__init__()

        self.transition = nn.Sequential(
            nn.Linear(in_features, out_features),
            nn.Mish(),
        )

    def forward(self, x):
        return self.transition(x)      

      
# ------------------------
# Пример модели с 4 слоями
# ------------------------

def gaussian_rbf(distances):
    return torch.exp(-distances)


class DenseMRBFN(nn.Module):
    def __init__(self, in_features, hidden_units1, hidden_units2, hidden_units3, out_features, per_dimension_sigma=False):
        super(DenseMRBFN, self).__init__()

        self.rbf_block1 = RBFDenseBlock(in_features, hidden_units1, gaussian_rbf, per_dimension_sigma)
        self.transition1 = TransitionLayer(in_features + hidden_units1, hidden_units1)

        self.rbf_block2 = RBFDenseBlock(in_features + hidden_units1, hidden_units2, gaussian_rbf, per_dimension_sigma)
        self.transition2 = TransitionLayer(in_features + hidden_units1 + hidden_units2, hidden_units2)

        self.rbf_block3 = RBFDenseBlock(in_features + hidden_units1 + hidden_units2, hidden_units3, gaussian_rbf, per_dimension_sigma)
        self.transition3 = TransitionLayer(in_features + hidden_units1 + hidden_units2 + hidden_units3, hidden_units3)

        self.linear_final = nn.Linear(hidden_units3, out_features)

    def forward(self, x):
        outputs = [x]

        x1 = self.rbf_block1(x, outputs)
        outputs.append(x1)
        x1 = self.transition1(torch.cat(outputs, dim=1))

        x2 = self.rbf_block2(x1, outputs)
        outputs.append(x2)
        x2 = self.transition2(torch.cat(outputs, dim=1))

        x3 = self.rbf_block3(x2, outputs)
        outputs.append(x3)
        x3 = self.transition3(torch.cat(outputs, dim=1))

        return self.linear_final(x3)

4. Объяснение архитектуры KAN: Kolmogorov-Arnold Networks

Для начала предлагаю определить ключевые термины в контексте данного раздела, чтобы избежать путаницы в дальнейшем.

KAN – семейство нейронных сетей прямого распространения имеющих следующий вид:

${\small f_\text{KAN}(\mathbf{x}) = \sum_{i_L=1}^{n_L} \phi_{L-1, i_L, i_{L-1}} \left( \sum_{i_{L-1}=1}^{n_{L-1}} \phi_{L-2, i_{L-1}, i_{L-2}} \left( \cdots \left( \sum_{i_1=1}^{n_1} \phi_{1, i_2, i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0, i_1, i_0}(x_{i_0}) \right) \right) \right) \right)}$

То есть искомую функцию можно разложить на суперпозиции суммы функций одной переменной $\phi_{l,i_{l+1},i_l}(x_{i_l}).$ В последнем слое индекс i_L означает, что функция может быть векторозначной.

Где каждая сумма (слой) $\sum_{i_l=1}^{n_l} \phi_{l,i_{l+1},i_l}(x_{i_l})$ представляет собой преобразование подобного рода:

$\mathbf{x}_{l+1} = \sum_{i_l=1}^{n_l} \phi_{l,i_{l+1},i_l}(x_{i_l}) \\[20pt] \\= \left[ \begin{array}{cccc} \phi_{l,1,1}(x_{1}) & + & \phi_{l,1,2}(x_{2}) & + \cdots + & \phi_{l,1,n_l}(x_{n_l}) \\ \phi_{l,2,1}(x_{1}) & + & \phi_{l,2,2}(x_{2}) & + \cdots + & \phi_{l,2,n_l}(x_{n_l}) \\ \vdots & & \vdots & \ddots & \vdots \\ \phi_{l,n_{l+1},1}(x_{1}) & + & \phi_{l,n_{l+1},2}(x_{2}) & + \cdots + & \phi_{l,n_{l+1},n_l}(x_{n_l}) \\ \end{array} \right]$

B-spline KAN – частный случай KAN при $\phi_{l,i_{l+1},i_l}(x_{i_l}) = \phi_{\text{BSKAN }l, i_{l+1}, i_l}(x_{i_l})$

$\phi_\text{BSKAN}(x)=w_bf_\text{base}(x)+w_s f_\text{spline}(x)$ $f_\text{base} = \text{silu}(x) = \frac{x}{1 + e^{-x}}$ $f_\text{spline}(x) = \sum_i w_i B^k_i(x), \\[10pt] \text{где}\ B_i(x) \ \text{– B-сплайн степени k на одном слое}$

Нейронная сеть будет иметь следующий вид:

$f_\text{BSKAN}(\mathbf{x}) = \sum_{i_L=1}^{n_L} \phi_{\scriptscriptstyle \text{BSKAN } L-1, i_L, i_{L-1}} \left( \cdots \left( \sum_{i_1=1}^{n_1} \phi_{\scriptscriptstyle \text{BSKAN } 1, i_2, i_1} \left( \sum_{i_0=1}^{n_0} \phi_{\scriptscriptstyle \text{BSKAN } 0, i_1, i_0} (x_{i_0}) \right) \right) \right)$

Где каждый слой представляет собой преобразование подобного рода:

$\mathbf{x}_{l+1} = \sum_{i_l=1}^{n_l} \phi_{\text{BSKAN }l, i_{l+1}, i_l}(x_{i_l}) \\[20pt]= \begin{bmatrix} \phi_{\text{BSKAN }_{l,1,1}}(x_{1}) + \phi_{\text{BSKAN }_{l,1,2}}(x_{2}) + \dots + \phi_{\text{BSKAN }_{l,1,n_l}}(x_{n_l}) \\ \phi_{\text{BSKAN }_{l,2,1}}(x_{1}) + \phi_{\text{BSKAN }_{l,2,2}}(x_{2}) + \dots + \phi_{\text{BSKAN }_{l,2,n_l}}(x_{n_l}) \\ \vdots \\ \phi_{\text{BSKAN }_{l,n_{l+1},1}}(x_{1}) + \phi_{\text{BSKAN }_{l,n_{l+1},2}}(x_{2}) + \dots + \phi_{\text{BSKAN }_{l,n_{l+1},n_l}}(x_{n_l}) \end{bmatrix}$

Официальная реализация B-spline KAN доступна в репозитории pykan на GitHub.

Хочу также уточнить, что авторы оригинального исследования определяют также KAN, но остановились на B-сплайнах как удобном выборе для параметризации этих одномерных функций. Они просто рассматривают реализацию, тогда как моя цель – обобщить многие архитектуры, используя их расширение теоремы Колмогорова-Арнольда на произвольный случай. Поэтому я и назвал их реализацию B-spline KAN, отделив её от KAN.

4.1 MLP как частный случай KAN

Мы уже рассматривали аффинное преобразование и применение функции активации как суперпозицию функций одной переменной в пункте: 2.2 «Аффинное преобразование и применение функции активации как функции преобразования и суперпозиции суммы функций».

$\mathbf{x}_{l+1} = \sum_{i_l=1}^{n_l} \phi_{l,i_{l+1},i_l}(x_{i_l}) \\[20pt] =\left[ \begin{array}{cccc} \phi_{l,1,1}(x_{1}) & + & \phi_{l,1,2}(x_{2}) & + \cdots + & \phi_{l,1,n_l}(x_{n_l}) \\ \phi_{l,2,1}(x_{1}) & + & \phi_{l,2,2}(x_{2}) & + \cdots + & \phi_{l,2,n_l}(x_{n_l}) \\ \vdots & & \vdots & \ddots & \vdots \\ \phi_{l,n_{l+1},1}(x_{1}) & + & \phi_{l,n_{l+1},2}(x_{2}) & + \cdots + & \phi_{l,n_{l+1},n_l}(x_{n_l}) \\ \end{array} \right], \\[10pt] \quad \\[10pt] \text{где} \\[10pt] \phi_{l,i_{l+1},i_l}(x_{i_l}) = w_{l,i_{l+1},i_l }\cdot f_{\text{activation, }l}(x_{i_l})+ b_{l,i_{l+1}, i_l}$

Если мы сравним это со слоем в KAN, то увидим, что функция активации и последующее аффинное преобразование составляют один слой в KAN. Первый слой, очевидно, будет состоять из линейных функций. $\phi_{0,i_1,i_0}(x_{l,i}) = w_{ i_1, i_0} \cdot x_{i_0}+ b_{ i_1, i_0}.$ И в контексте данной интерпретации все функции $\phi_{l,i_{l+1},i_l}(x_{i_l})$ являются обучаемыми, хоть и не могут аппроксимировать в большинстве своем функции одной переменной. Это связано с тем, что они, по сути, являются масштабированными функциями активации, которые также в большинстве своем не обладают данной возможностью.

Тем самым можно заключить логичный вывод, что MLP является частным случаем KAN. Несмотря на то, что KAN и MLP являются подвидами нейронных сетей прямого распространения, KAN обобщает MLP на случай произвольных функций одной переменной.

Но также предлагаю рассмотреть, что из себя представляет часть существующих теорем универсальной аппроксимации для MLP в контексте KAN. Для начала рассмотрим теорему Цыбенко, которая утверждает, что нейронные сети с одним скрытым слоем и произвольным числом нейронов, использующие сигмоидные функции активации, обладают свойством универсальной аппроксимации.

Это идентично KAN с двумя слоями:

$f_\text{Cybenko}(\mathbf{x}) = \sum_{i_1=1}^{n_1} \phi_{1,i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0,i_1,i_0}(x_{i_0}) \right)$

В первом слое все функции будут линейными, а во втором используются сигмоидные функции активации и масштабирующие коэффициенты для каждой из них.

Первый слой:

$\mathbf{x}_1 = \sum_{i_0=1}^{n_0} w_{i_1,i_0} x_{i_0} + b_{i_1} \\[20pt] =\begin{bmatrix} w_{1,1} x_{1} & + & w_{1,2} x_{2} & + & \cdots & + & w_{1,n_0} x_{n_0} & + & b_{1} \\ w_{2,1} x_{1} & + & w_{2,2} x_{2} & + & \cdots & + & w_{2,n_0} x_{n_0} & + & b_{2} \\ \vdots & & \vdots & & \ddots & & \vdots & & \vdots \\ w_{n_1,1} x_{1} & + & w_{n_1,2} x_{2} & + & \cdots & + & w_{n_1,n_0} x_{n_0} & + & b_{n_1} \end{bmatrix}\\[30pt] = \sum_{i_0=1}^{n_0} \phi_{i_1,i_0}(x_{i_0})\\[20pt] = \begin{bmatrix} \phi_{1,1}(x_{1}) + \phi_{1,2}(x_{2}) + \dots + \phi_{1,n_0}(x_{n_0}) + \phi_{1,n_0}(x_{n_0}) \\[20pt] \phi_{2,1}(x_{1}) + \phi_{2,2}(x_{2}) + \dots + \phi_{2,n_0}(x_{n_0}) + \phi_{2,n_0}(x_{n_0}) \\ \vdots \\ \phi_{n_1,1}(x_{1}) + \phi_{n_1,2}(x_{2}) + \dots + \phi_{n_1,n_0}(x_{n_0}) + \phi_{n_1,n_0}(x_{n_0}) \end{bmatrix}, \\[50pt]$

И второй выходной:

$f_\text{Cybenko}(\mathbf{x}_1) = \sum_{i_1=1}^{n_1} w_{i_1} \sigma(x_{i_!}) +b \\[25pt] = w_{1} \sigma(x_1) + w_{2} \sigma(x_2) + \cdots + w_{n_1} \sigma(x_{n_1}) +b\\[20pt]= \sum_{i_1=1}^{n_1} \phi_{1,i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0,i_1,i_0}(x_{i_0}) \right)$

В итоге мы получаем модель, в которой сначала выполняется аффинное преобразование в определённое подпространство- мерного пространства, затем применяется сигмоидная функция активации, после чего выполняется ещё одно аффинное преобразование и на выходе получается скаляр (для векторнозначных функций – вектор).

$f_\text{Cybenko}(\mathbf{x}) = \mathbf{W}_2 \cdot \sigma(\mathbf{W}_1 \cdot \mathbf{x}+\mathbf{b}_\text{in})+\mathbf{b}_\text{out}$

И в контексте KAN, теорему универсальной аппроксимации Цыбенко можно рассмотреть как теорему, которая определяет структуру KAN, которая обладает свойствами универсальной аппроксимации.

Также предлагаю рассмотреть случай ограниченной ширины и глубины нейронной сети, взяв к примеру теорему универсальной аппроксимации Н. Кулиева и В. Исмаилова из данной работы.

Котоорая гласит, что любая непрерывная функция, определённая на -мерной области, может быть аппроксимирована нейронной сетью с двумя скрытыми слоями и сигмоидными и функциями активации:

$f(x)_\text{Guliyev, Ismailov} = \sum_{p=1}^{2d+2} w_p \, \sigma\left( \sum_{q=1}^d w_{pq} \, \sigma\left( x_q + b_{pq} \right) + b_p \right)$

Первый слой:

$\mathbf{x}_1 = \sum_{q=1}^d \phi_{0,p,q}(x_q) \\[20pt]= \begin{bmatrix} w_{1,1}\sigma(w_{1,1}x_1 - b_{1,1}) & + & \dots & + & w_{1,d}\sigma(w_{1,d}x_d - b_{1,d}) \\ w_{2,1}\sigma(w_{2,1}x_1 - b_{2,1}) & + & \dots & + & w_{2,d}\sigma(w_{2,d}x_d - b_{2,d}) \\ \vdots & & \ddots & & \vdots \\ w_{2d+2,1}\sigma(w_{2d+2,1}x_1 - b_{2d+2,1}) & + & \dots & + & w_{2d+2,d}\sigma(w_{2d+2,d}x_d - b_{2d+2,d}) \\ \end{bmatrix}$

Второй слой (выходной):

$f(x)_\text{Guliyev, Ismailov} = \sum_{p=1}^{2d+2} \phi_{1,p}(x_p) \\[20pt] = \begin{bmatrix} w_{1}\sigma(\cdot) + \dots + w_{2d+2}\sigma(\cdot) \end{bmatrix} \cdot \begin{bmatrix} \sum_{q=1}^{d} w_{1,q}\sigma(w_{1,q}x_q - b_{1,q}) - b_{1} \\ \sum_{q=1}^{d} w_{2,q}\sigma(w_{2,q}x_q - b_{2,q}) - b_{2} \\ \vdots \\ \sum_{q=1}^{d} w_{2d+2,q}\sigma(w_{2d+2,q}x_q - b_{2d+2,q}) - b_{2d+2} \end{bmatrix}$

Собственно также, как и в случае с теоремой Цыбенко, здесь мы определяем структуру KAN с ограниченной шириной и глубиной, которая обладает универсальной аппроксимацией.

Ну и рассмотрим теоремы универсальной аппроксимации для случая произвольной ширины и глубины, которые упоминались в начале раздела про MLP, а именно теоремы Хорника и Лешно. Если рассматривать эти случаи в контексте KAN, то первый слой представляет собой аффинное преобразование, а последующие слои – преобразования, состоящие из масштабированных функций активации. При этом функция активации одинакова для всего слоя. В случае Хорника она должна быть непрерывной, ограниченной, неконстантой и неполиномиальной, а в случае теоремы Лешно допускаются кусочно-непрерывные функции и лишь локально ограниченные, а также не являющиеся полиномами почти всюду.

${\small f(\mathbf{x}) = \sum_{i_L=1}^{n_L} \phi_{L-1, i_L, i_{L-1}} \left( \sum_{i_{L-1}=1}^{n_{L-1}} \phi_{L-2, i_{L-1}, i_{L-2}} \left( \cdots \left( \sum_{i_1=1}^{n_1} \phi_{1, i_2, i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0, i_1, i_0}(x_{i_0}) \right) \right) \cdots \right) \right)}$

Где во входном слое фукнции $\phi_{0,i_1,i_0}(x_{l,i}) = w_{ i_1, i_0}\cdot x_{i_0} + b_{ i_1, i_0},$ а во всех остальных – $\phi_{l,i_{l+1},i_l}(x_{i_l}) = w_{l,i_{l+1}, i_l}\cdot f_{\text{activation, }l}(x_{i_l})+ b_{l, i_{l+1}, i_l}.$

4.2 Теорема Колмогорова — Арнольда

Исследователи из MIT, предложив архитектуру B-spline KAN, вдохновились теоремой Колмогорова-Арнольда о представлении функций и расширили её. Однако в данном случае я предлагаю рассмотреть не архитектуру KAN через призму теоремы Колмогорова-Арнольда, а наоборот, что представляет собой теорема Колмогорова-Арнольда в контексте архитектуры KAN.

Андрей Колмогоров и Владимир Арнольд установили, что еслиявляется функцией нескольких перменных, то может быть записана как конечная композиция непрерывных функций одной переменной и операций сложения:

$f(\mathbf{x}) = f(x_1, \ldots, x_n) = \sum_{q=0}^{2n} \Phi_q \left( \sum_{p=1}^n \phi_{q,p}(x_p) \right),\\[20pt]\text{где } \phi_{q,p} : [0, 1] \to \mathbb{R} \text{ и } \Phi_q : \mathbb{R} \to \mathbb{R}.$

И хотя это не теорема универсальной аппроксимации, так как она утверждает просто о способности разложения функции, в контексте архитектуры KAN теорему Колмогорова-Арнольда можно рассматривать как теорему, определяющую структуру сети KAN и образующую подмножество сетей с ограниченной шириной и глубиной.

Однако можно рассмотреть её вариант, который очевидно является элементом множества KAN и, в том числе, элементом подмножества KAN, в виде MLP. Мы уже разбирали его в предыдущем пункте, а именно – теорему универсальной аппроксимации Н. Кулиева и В. Исмаилова для двухслойного MLP. В качестве основы для доказательства они использовали теорему Колмогорова-Арнольда, по сути доказывая, что частный случай теоремы способен аппроксимировать любую непрерывную функцию на компактном множестве, с контролируемой ошибкой аппроксимации.

При этом интересно, что Теорема Колмогорова-Арнольда появилась задолго до известных архитектур 1990-х годов. Более того, она была опубликована в тот же год, когда Фрэнк Розенблатт предложил идею перцептрона – в 1957 году. И только спустя 67 лет исследователи из MIT, вдохновившись этой теоремой, предложили идею для создания KAN, который объединяет огромное количество вариантов сетей прямого распространения.

4.3 B-spline KAT и B-spline KAN

Рассмотрим для начала B-spline теорему Колмогорова-Арнольда (B-spline KAT). Мы уже обсуждали, что можем представить MLP в контексте KAN как нейронную сеть с первым слоем аффинного преобразования, где каждая функция имеет вид $\phi(x)= w\cdot x+b,$ и последующими преобразованиями, где функция в общем виде $\phi(x) = w\cdot f_{\text{activation}}(x) +b,$ в которой $f_{\text{activation}}(x)$ заранее определена для всего слоя.

Несмотря на то, что функции $\phi(x)$ являются обучаемыми, при использовании популярных функций активации, к примеру ReLU, Mish, Swish (SiLU), GELU, функции по сути могут аппроксимировать с заданной точностью крайне маленький класс функций. Поэтому мы можем переопределить KAN, заменив все функции на произвольные B-сплайны, которые обладают куда большей возможностью аппроксимирования, что потенциально может дать больше гибкости модели, а также избавит от проблем выбора конкретной функции активации. Собственно, данный подход и предложили исследователи из MIT как вариант реализации KAN, а также выдвинули и доказали теоретическую часть для этого:

Теорема утверждает, что если целевая функция f(x) представлена как последовательность преобразований:

$f(x) = (\Phi_{L-1} \circ \Phi_{L-2} \circ \dots \circ \Phi_1 \circ \Phi_0)x,$

где каждое $\Phi_{l,i_{l+1},i_l}$ – это непрерывно дифференцируемая функция (k+1) - раз.
То её можно эффективно аппроксимировать с помощью B-сплайнов.

При этом точность аппроксимации будет зависеть от параметра который представляет шаг сетки аппроксимации. Для любых $m \in [0, k],$ ошибка аппроксимации f(x) через B-сплайны в C^m - норме будет оцениваться rак:

$\| f(x) - f_G(x) \|_{C^m} \leq C G^{-k+1+m}$

где:

– шаг сетки (чем меньше, тем точнее будет аппроксимация),
– константа, зависящая от функции и её преобразований,
– степень гладкости исходных функций $\Phi$ ,
– порядок производной, до которого измеряется точность.

В контексте KAN данная теорема определяет структуру KAN и предоставляет аппроксимацию любой функции в пространстве непрерывных функций, обладающих непрерывными производными порядка k+1 на компактном множестве– то есть в $C^{k+1}(K).$ При этом множество $C^{k+1}(K)$ является подмножеством пространства непрерывных функций C(K).

И хотя в оригинальном исследовании B-spline KAT предлагаются как альтернатива традиционным теоремам универсальной аппроксимации, её природа аналогична, так как она также утверждает, что существует такая последовательность преобразований $\Phi_1, \Phi_2, \dots\ ,$ и что при достаточном количестве нейронов мы можем аппроксимировать любую функцию из определённого множества функций с заданной точностью в пределах $\epsilon.$ Таким образом, B-spline KAT можно также отнести к их числу!

Теперь перейдём к B-spline KAN. Вспомним, что в нем каждая из функций одной переменной задается как:

$\phi_\text{BSKAN}(x)=w_bf_\text{base}(x)+w_s{f_\text{spline}}(x)$ $f_\text{base} = \text{silu}(x) = \frac{x}{1 + e^{-x}}$ $f_\text{spline}(x) = \sum_i w_i B^k_i(x), \\[10pt] \text{где}\ B_i(x) \ \text{– B-сплайн степени k на одном слое}$

Можно, конечно, интерпретировать, что мы берем модель из B-spline KAT, состоящую из композиции сумм B-сплайнов, добавив просто взвешенную функцию активации от входного вектора. Однако по сути мы берем MLP с $\phi(x) = w\cdot f_{\text{activation}}(x) +b,$ где вместо константной функции в виде смещения $f_\text{bias}(x) = b,$ мы используем B-сплайны $f_\text{bias} = w_s{f_{spline}}(x).$ Очевидно, что сплайны могут без проблем аппроксимировать на компактном множестве константную функцию, а значит, теоретически данная модель может сойтись либо к чистому MLP, либо к модификации MLP. Это, в свою очередь, означает, что B-spline KAN в реализации, которую предложили в официальном репозитории pykan, больше напоминает модификацию MLP, чем альтернативную модель, предложенную в B-spline KAT.

4.4 SVM и RBF нейронная сеть как частный случай KAN

Мы уже знаем, что SVM можно представить как нейронную сеть прямого распространения и в некоторых случаях как MLP. В данном случае с линейным ядром, как в общем виде, так и с его разложением, у нас будет, по сути, два слоя с линейным и аффинным преобразованием.

$f_{\text{Linear SVM}}(\mathbf{x}) =\sum_{i=1}^{n} \alpha_i y_i K_\text{Linear}(\mathbf{x}_\text{train}, \mathbf{x}) + b = \mathbf{W}_2(\mathbf{W}_1 \cdot \varphi_\text{linear}(\mathbf{x})) + b \\= \sum_{i=1}^{n} \alpha_i y_i (\mathbf{x}_\text{train}\cdot \mathbf{x}) + b$

Если представить в формате KAN, то это будет выглядеть следующим образом:

$f_\text{Linear SVM}(x) = \sum_{i_1=1}^{n_1} \phi_{1,i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0,i_1,i_0}(x_{i_0}) \right),\\[10pt] \text{где} \\[10pt] \phi_{0,i_1,i_0}(x_{i_0}) = x_{\text{train},i_1, i_0} \cdot x_{i_0}, \\[10pt] \phi_{1,i_1}(x_{i_1}) = \alpha_{i_1} \cdot y_{i_1} \cdot x_{i_1} +b_{i_1}$

С полиномиальным ядром всё в целом аналогично: если не использовать разложение с помощью функции $\varphi_{\text{poly }n},$ то мы просто изменяем выходной слой и добавляем смещение во входной слой, производя аффинное преобразование, а не линейное.

$f_\text{Poly SVM}(\mathbf{x}) = \sum_{i=1}^{n} \alpha_i y_i \left( \mathbf{x}_\text{train} \cdot \mathbf{x} + \mathbf{b}_\text{in} \right)^k + b_\text{out} = \sum_{i_1=1}^{n_1} \phi_{1,i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0,i_1,i_0}({x}_{i_0}) \right) , \\ \text{где}\\[10pt] \phi_{0,i_1,i_0}({x}_{i_0}) = {x}_{\text{train},i_1, i_0} \cdot {x}_{i_0} + b_{\text{in}, i_1}, \\[10pt] \phi_{1,i_1}(x_{i_1}) = \alpha_{i_1} \cdot y_{i_1} \cdot \left( x_{i_1} \right)^k + b_\text{out}.$

А вот с разложением с помощью функции $\varphi_{\text{poly }n}$ всё посложнее. Мы знаем, что формула для SVM выглядит подобным образом (если $\mathbf{b}_\text{in}$ состоит из нулей):

$f_\text{Poly SVM}(\mathbf{x})_ = \mathbf{W}_2(\mathbf{W}_1 \cdot \varphi_{\text{poly }n}(\mathbf{x}))+ b_\text{out}$

Возьмем простой и привычный пример для двумерных входных данных:

$\varphi_{\text{poly }2}(\mathbf{x}) = \begin{bmatrix}x_1^2 \\x_2^2\\ \sqrt{2}x_1x_2\end{bmatrix}$

Нам было бы очень интересно подать в виде:

$\varphi_{\text{poly }2}(\mathbf{x}) = \begin{bmatrix} \phi_{1,1}(x_1) + \phi_{1,2}(x_2) \\ \phi_{2,1}(x_1) + \phi_{2,2}(x_2) \\ \phi_{3,1}(x_1) + \phi_{3,2}(x_2) \\ \end{bmatrix}$

Однако последний элемент в виде $\sqrt{2}x_1x_2$ всё портит. Безусловно, мы его можем разложить подобным образом:

$\sqrt{2}x_1x_2 = \frac{\sqrt{2}}2((x_1 + x_2)^2 - x_1^2-x_2^2)$

Но увы, данный вариант не будет работать для произвольного случая. К примеру, для $\varphi_{\text{poly }n}$ полиномиального ядра третьей степени степени для двумерного входного вектора появляется член – $\sqrt{3}x_1^2x_2,$ а его разложение будет выглядеть следующим образом:

$\sqrt{3}x_1^2x_2 = \frac{\sqrt{3}}3((x_1 + x_2)^3 - 3x_1x_2^2 -x_1^3-x_2^3)$

Как можно увидеть, тут не обойтись без функции двух переменных, которая также раскладывается и так бесконечное количество раз. Для произвольного случая все аналогично, и это можно вывести, используя мультиномиальную теорему.

Существует вариант разложить подобным образом:

$c \cdot \prod_{i=1}^n x_i^{a_i} = c \cdot \exp\left(\sum_{i=1}^n a_i \ln|x_i|)\right), x_i>0$

Но условие положительности, накладываемое логарифмом, мешает определить конечный знак данного произведения. Мы можем использовать знаковую функцию в виде комплексного члена. И тогда наше преобразование будет выглядеть следующим образом:

$c \cdot \prod_{i=1}^n x_i^{a_i} = c \cdot \exp\left( \sum_{i=1}^n a_i \left( \ln|x_i| + i \arg(x_i) \right) \right), x_i \neq0, \\[20pt] arg(x_i) - \text{фаза } x_i$

Исходя из этого всего, можно увидеть, что преобразование с помощью функции $\varphi_{\text{poly }n}(\mathbf{x})$ , что $\varphi_\text{RBF}(\mathbf{x})$ можно разложить на два слоя KAN.

Теперь рассмотрим RBF-нейронную сеть с одним скрытым слоем и произвольным количеством нейронов в контексте KAN, но уже с использованием ядерного трюка, как это предложено в теореме универсальной аппроксимации (Дж. Парк, В. Сандберг), которую мы обсуждали в пункте 3.2 «RBF нейронная сеть». В контексте KAN её можно воспринимать как теорему, которая формирует KAN для приближения широкого класса функций. Аналогично обсуждаемым ранее теоремам, её структура будет выглядеть следующим образом:

$f_\text{Park, Sandberg}(\mathbf{x}) = \sum_{i_1=1}^{n_1} \phi_{1,i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0,i_1,i_0}(x_{i_0}) \right) = \sum_{i=1}^{N} w_i K_\text{RBF}(\mathbf{x}, \mathbf{c}_i)$

Первый слой:
$\sum_{i_0=1}^{n_0} \phi_{0,i_1,i_0}(x_{i_0}) = \sum_{i_0=1}^{n_0} (x_{i_0} - c_{i_1,i_0})^2 \\[20pt] = {\small\left[ \begin{matrix} (x_{0,1} - c_{1,1})^2 & + & (x_{0,2} - c_{1,2})^2 & + & \dots & + & (x_{0,n_0} - c_{1,n_0})^2 \\ (x_{1,1} - c_{2,1})^2 & + & (x_{1,2} - c_{2,2})^2 & + & \dots & + & (x_{1,n_0} - c_{2,n_0})^2 \\ \vdots & & \vdots & & \ddots & & \vdots \\ (x_{n_1,1} - c_{n_1,1})^2 & + & (x_{n_1,2} - c_{n_1,2})^2 & + & \dots & + & (x_{n_1,n_0} - c_{n_1,n_0})^2 \\ \end{matrix} \right]}$
Второй слой:
- Гауссово ядро:
  $\sum_{i_1=1}^{n_1} \phi_{1, i_1}(x_{i_1}) = \sum_{i_1=1}^{n_1} w_{i_1} \cdot \exp\left(- \frac{(x_{i_1})^2}{\lambda_{i_1}}\right) \\[20pt] = w_{1} \cdot \exp\left(- \frac{(x_{1})^2}{\lambda_{1}}\right) + w_{2} \cdot \exp\left(- \frac{(x_{2})^2}{\lambda_{2}}\right) + \dots + w_{n_1} \cdot \exp\left(- \frac{(x_{n_1})^2}{\lambda_{n_1}}\right)$
- Обратное квадратичное ядро (Inverse Quadratic):
  $\sum_{i_1=1}^{n_1} \phi_{1,i_1}(x_{i_1}) = \sum_{i_1=1}^{n_1} w_{i,i_1} \cdot \frac{1}{1 + \left(\lambda_{i_1} \cdot x_{i_1}\right)^2} \\[20pt] = w_{1} \cdot \frac{1}{1 + \left(\lambda_{1} \cdot x_{1}\right)^2} + w_{2} \cdot \frac{1}{1 + \left(\lambda_{2} \cdot x_{2}\right)^2} + \cdots + w_{n_1} \cdot \frac{1}{1 + \left(\lambda_{n_1} \cdot x_{n_1}\right)^2}$
- Обратное мультиквадратичное ядро (Inverse Multiquadric):
  $\sum_{i_1=1}^{n_1} \phi_{1, i_1}(x_{i_1}) = \sum_{i_1=1}^{n_1} w_{i_1} \cdot \frac{1}{\sqrt{1 + \lambda_{i_1} \cdot (x_{i_1})^2}} \\[20pt] = w_{1} \cdot \frac{1}{\sqrt{1 + \lambda_{1} \cdot (x_{1})^2}} + w_{2} \cdot \frac{1}{\sqrt{1 + \lambda_{2} \cdot (x_{2})^2}} + \dots + w_{n_1} \cdot \frac{1}{\sqrt{1 + \lambda_{n_1} \cdot (x_{n_1})^2}}$

Для теоремы универсальной аппроксимации RBF-нейронной сети с произвольным количеством скрытых слоев и нейронов, которую я доказывал в пункте 3.2.1 «Многослойная RBF нейронная сеть», всё аналогично. Она также формирует KAN и выглядит всё так:

Для
$\sum_{i_0=1}^{n_0} \phi_{0,i_1,i_0}(x_{i_0}) = \sum_{i_0=1}^{n_0} (x_{i_0} - c_{i_1,i_0})^2 \\[30pt] = {\small \left[ \begin{matrix} (x_{0,1} - c_{1,1})^2 & + & (x_{0,2} - c_{1,2})^2 & + & \dots & + & (x_{0,n_0} - c_{1,n_0})^2 \\ (x_{1,1} - c_{2,1})^2 & + & (x_{1,2} - c_{2,2})^2 & + & \dots & + & (x_{1,n_0} - c_{2,n_0})^2 \\ \vdots & & \vdots & & \ddots & & \vdots \\ (x_{n_1,1} - c_{n_1,1})^2 & + & (x_{n_1,2} - c_{n_1,2})^2 & + & \dots & + & (x_{n_1,n_0} - c_{n_1,n_0})^2 \\ \end{matrix} \right]}\\[40pt]$
Для $1\leq l\leq L-2$
$\sum_{i_l=1}^{n_l} \phi_{i_l,i_{l+1},i_l}(x_{i_l}) = \sum_{i_l=1}^{n_l} (c_{i_{l+1}, i_l} - \exp\left(- \frac{(x_{i_l})^2}{\lambda_{i_{l+1}}}\right))^2 \\ = \left[ \begin{matrix} (c_{1,1} - \exp\left(- \frac{(x_{1,1})^2}{\lambda_1}\right))^2 + \dots + (c_{1, n_l} - \exp\left(- \frac{(x_{1, n_l})^2}{\lambda_1}\right))^2 \\ (c_{2,1} - \exp\left(- \frac{(x_{2,1})^2}{\lambda_2}\right))^2 + \dots + (c_{2, n_l} - \exp\left(- \frac{(x_{2, n_l})^2}{\lambda_2}\right))^2 \\ \vdots \\ (c_{n_{l+1},1} - \exp\left(- \frac{(x_{n_{l+1},1})^2}{\lambda_{n_{l+1}}}\right))^2 + \dots + (c_{n_{l+1}, n_l} - \exp\left(- \frac{(x_{n_{l+1}, n_l})^2}{\lambda_{n_{l+1}}}\right))^2 \\ \end{matrix} \right]\\[60pt]$
Для
$\sum_{i_{L-1}=1}^{n_{L-1}} \phi_{L-1, i_{L}, i_{L-1}}(x_{i_{L-1}}) = \sum_{i_{L-1}=1}^{n_{L-1}} w_{i_{L}, i_{L-1}} \cdot \exp\left(- \frac{(x_{i_{L-1}})^2}{\lambda_{i_{L-1}}}\right) \\[20pt] = \left[ \begin{matrix} w_{1,1} \cdot \exp\left(- \frac{(x_{1})^2}{\lambda_{1}}\right) + w_{1,2} \cdot \exp\left(- \frac{(x_{2})^2}{\lambda_{2}}\right) + \dots + w_{1,n_{L-1}} \cdot \exp\left(- \frac{(x_{n_{L-1}})^2}{\lambda_{n_{L-1}}}\right) \\ \vdots \end{matrix} \right]\\[30pt]$

4.4.1 Многослойная RBF нейронная сеть (MRBFN) vs FastKAN

Вскоре после выхода исследования про KAN была выпущена модель, которая вместо использования B-спланов использует Гауссово ядро. Сама модель доступна на GitHub, а также её описание на arXiv.

Цель данного сравнения – показать, что, несмотря на то, что с первого взгляда мы получаем KAN с RBF-ядрами, который быстрее B-spline KAN и по точности ± такой же, и как бы всё отлично, но у нас уже существует подобный KAN в виде многослойной RBF-нейронной сети, который ничем не хуже.

Архитектура FastKAN выглядит следующим образом:

$f_\text{FastKAN}(\mathbf{x}) = {\small \sum_{i_L=1}^{n_L} \phi_{L-1, i_L, i_{L-1}} \left( \sum_{i_{L-1}=1}^{n_{L-1}} \phi_{L-2, i_{L-1}, i_{L-2}} \left( \cdots \left( \sum_{i_1=1}^{n_1} \phi_{1, i_2, i_1} \left( \sum_{i_0=1}^{n_0} \phi_{0, i_1, i_0}(x_{i_0}) \right) \right) \right) \right)}\\[20pt]где \\[10pt] \phi_{l,i_{l+1},i_l}(x_{l,i_l}). = K_\text{RBF}(c_{i_{l+1},i_l}, x_{l, i_l})$

Для ${0\leq l\leq L-2}$
$\sum_{i_{l}=1}^{n_{l}} \phi_{l, i_{l+1}, i_{l}}(x_{i_{l}}) = \sum_{i_{l}=1}^{n_{l}} K_\text{RBF}(c_{i_{l+1}, i_{l}}, x_{i_{l}}) \\[20pt] = \left[ \begin{matrix} K_\text{RBF}(c_{1,1}, x_{1}) + K_\text{RBF}(c_{1,2}, x_{2}) + \dots + K_\text{RBF}(c_{1, n_{l}}, x_{n_{l}}) \\ K_\text{RBF}(c_{2,1}, x_{1}) + K_\text{RBF}(c_{2,2}, x_{2}) + \dots + K_\text{RBF}(c_{2, n_{l}}, x_{n_{l}}) \\ \vdots \\ K_\text{RBF}(c_{n_{l+1},1}, x_{1}) + K_\text{RBF}(c_{n_{l+1},2}, x_{2}) + \dots + K_\text{RBF}(c_{n_{l+1}, n_{l}}, x_{n_{l}}) \\ \end{matrix} \right] \\[40pt]$
Для
$\sum_{i_{L-1}=1}^{n_{L-1}} \phi_{L-1, i_{L}, i_{L-1}}(x_{i_{L-1}}) = \sum_{i_{L-2}=1}^{n_{L-2}} K_\text{RBF}(c_{i_{L}, i_{L-1}}, x_{i_{L-1}}) \\[20pt] = \left[ \begin{matrix} K_\text{RBF}(c_{1,1}, x_{1}) + K_\text{RBF}(c_{1,2}, x_{2}) + \dots + K_\text{RBF}(c_{1, n_{L-1}}, x_{n_{L-1}}) \\ \vdots \end{matrix} \right]\\[10pt]$

Из пункта 3.2 мы знаем, что преобразование с помощью слоя RBF-нейронной сети (кроме выходного) выглядит следующим образом:

$\mathbf{x}_{\text{output}} = \begin{bmatrix} K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_1) \\ K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_2) \\ K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_3) \\ \vdots \\ K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_n) \end{bmatrix}$

То есть разница в том, что в случае RBF-нейронной сети мы применяем Гауссово ядро между всем входным вектором и вектором веса, когда в FastKAN мы применяем между каждым элементом входного и вектора веса, а потом суммируем.

Для выполнения поставленной задачи нужно выполнить, по сути, данное сравнение в плане скорости и сложности преобразования.

$\begin{bmatrix} K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_1) \\ K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_2) \\ K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_3) \\ \vdots \\ K_{\text{RBF}}(\mathbf{x}_{\text{input}}, \mathbf{c}_n) \end{bmatrix} \quad \text{vs} \quad \begin{bmatrix} \sum_{q=1}^{M} K_{\text{RBF}}(\mathbf{x}_{\text{input}, q}, \mathbf{c}_{1, q}) \\ \sum_{q=1}^{M} K_{\text{RBF}}(\mathbf{x}_{\text{input}, q}, \mathbf{c}_{2, q}) \\ \sum_{q=1}^{M} K_{\text{RBF}}(\mathbf{x}_{\text{input}, q}, \mathbf{c}_{3, q}) \\ \vdots \\ \sum_{q=1}^{M} K_{\text{RBF}}(\mathbf{x}_{\text{input}, q}, \mathbf{c}_{n, q}) \end{bmatrix}$

По сути, нам достаточно сравнить преобразование, которое приводит в обоих случаях к элементу результирующего вектора. Это без проблем обобщается на весь слой и всю модель, то есть:

$K_\text{RBF}(\mathbf{x}, \mathbf{c}) \quad \text{vs} \quad \sum_{q=1}^{M} K_\text{RBF}(\mathbf{x}_q, \mathbf{c}_q) \\\\ \varphi_{\text{RBF}}(\mathbf{x})^\top \cdot \varphi_{\text{RBF}}(\mathbf{c}) \quad \text{vs} \quad \sum_{q=1}^{M} \varphi_{\text{RBF}}(\mathbf{x}_q)^\top \cdot \varphi_{\text{RBF}}(\mathbf{c}_q)$

Функции $\varphi_{\text{RBF}}$ для входных векторов размерностью больше двух в итоге дадут идентичный логический вывод из-за свойств функции. Поэтому рассмотрим на привычных двумерных векторах $\mathbf{x} = [x_1, x_2]$ и $\mathbf{с} = [с_1, с_2].$ Одномерный случай мы проанализируем после. Также в качестве примера ядра я буду использовать Гауссово. Для обратного квадратического и обратного мультиквадратического вывод идентичен, или для ядер, которые обладают теми же свойствами. Также сумма мультииндексов ограничена M =2, и для удобства $\lambda = 2.$

Начнем с:

$\sum_{q=1}^{m} K_\text{Gaussian}(x_q, с_q)$ $\varphi_\text{gaussian}(x_1) = \exp\left( -\frac{1}{2} x_1^2 \right) \left[ 1, x_1, \frac{x_1^2}{\sqrt{2}}, \dots \right]$ $\varphi_\text{gaussian}(x_2) = \exp\left( -\frac{1}{2} x_2^2 \right) \left[ 1, x_2, \frac{x_2^2}{\sqrt{2}}, \dots \right]$ $\varphi_\text{gaussian}(c_1) = \exp\left( -\frac{1}{2} c_1^2 \right) \left[ 1, c_1, \frac{c_1^2}{\sqrt{2}}, \dots \right]$ $\varphi_\text{gaussian}(c_1) = \exp\left( -\frac{1}{2} c_1^2 \right) \left[ 1, c_1, \frac{c_1^2}{\sqrt{2}}, \dots \right]$ $\sum_{q=1}^{2} K_{\text{Gaussian}}(x_q, c_q) = \sum_{q=1}^{2} \varphi_{\text{gaussian}}(x_q)^\top \cdot \varphi_{\text{gaussian}}(c_q) \\=\exp\left( -\frac{1}{2} (x_1^2 + c_1^2) \right) \left( 1 + x_1 c_1 + \frac{x_1^2 c_1^2}{2} + \dots \right) \\+\exp\left( -\frac{1}{2} (x_2^2 + c_2^2) \right) \left( 1 + x_2 c_2 + \frac{x_2^2 c_2^2}{2} + \dots \right)$

И теперь перейдем к $K_\text{Gaussian}(\mathbf{x}, \mathbf{с})$

$\varphi_\text{gaussian}(\mathbf{x}) = \exp\left( -\frac{1}{2} (x_1^2 + x_2^2) \right) \left[ 1, x_1, x_2, \frac{x_1^2}{\sqrt{2}}, x_1 x_2, \frac{x_2^2}{\sqrt{2}}, \dots \right]$ $\varphi_\text{gaussian}(\mathbf{c}) = \exp\left( -\frac{1}{2} (c_1^2 + c_2^2) \right) \left[ 1, c_1, c_2, \frac{c_1^2}{\sqrt{2}}, c_1 c_2, \frac{c_2^2}{\sqrt{2}}, \dots \right]$ $K_\text{Gaussian}(\mathbf{x}, \mathbf{c}) = \varphi_\text{gaussian}(\mathbf{x})^\top \cdot \varphi_\text{gaussian}(\mathbf{c}) \\[20pt]=\textstyle \exp\left( -\frac{1}{2} (x_1^2 + x_2^2 + c_1^2 + c_2^2) \right) \left( 1 + x_1 c_1 + x_2 c_2 + \frac{x_1^2 c_1^2}{2} + x_1 x_2 c_1 c_2 + \frac{x_2^2 c_2^2}{2} + \dots \right)$

Как мы можем заметить, разница в том, что для вычисления ядра между векторами у нас экспоненциальный множитель учитывает все элементы векторов, а также появляются смешанные члены вроде x_1 x_2 с_1 с_2.

Поэтому, выразим $K_\text{Gaussian}(\mathbf{x}, \mathbf{с})$ с помощью:

$\sum_{q=1}^{m} K_\text{Gaussian}(x_q, с_q)$ $K_\text{Gaussian}(\mathbf{x}, \mathbf{c}) = \varphi_\text{gaussian}(\mathbf{x})^\top \cdot \varphi_\text{gaussian}(\mathbf{c}) \\ =\exp\left( -\frac{1}{2} \sum_{q=1}^{M} (x_q^2 + c_q^2) \right) \cdot \left(\sum_{q=1}^{M} \varphi_{\text{Gaussian}}\left(\frac{x_q}{\exp(-0.5 x_q^2)}\right) \cdot \varphi_{\text{Gaussian}}\left(\frac{c_q}{\exp(-0.5 c_q^2)}\right)+ \\+ \sum_{\text{(смешанные члены)}} \right)$

Как мы можем увидеть, каждый элемент преобразования $\varphi_{\text{gaussian}}(\mathbf{x})^\top \cdot \varphi_{\text{gaussian}}(\mathbf{с})$ учитывает не просто экспоненту от суммы конкретных квадратов двух элементов, а от всех элементов двух векторов. Также мы учитываем совместные члены, которые учитывают множитель экспоненты от всех членов двух векторов. В совокупности это даёт очевидно больше возможностей для учета взаимосвязи между входными признаками, а значит, потенциально более высокую точность в задачах.

В контексте KAN в случае с RBF-нейронной сетью мы по сути имеем дополнительные слои при представлении сети с использованием функции $\varphi_\text{RBF}.$ Как мы обсуждали, можно разложить, используя экспоненту от суммы логарифмов с контролем знака, используя фазу данного числа в комплексной плоскости, что даёт дополнительный слой, в то время как в FastKAN подобных дополнительных слоев нет.

Теперь перейдём к скорости. Довольно очевидно, что для входных векторов с размерностью больше или равной двум RBF-нейронная сеть с использованием ядерного трюка будет всегда быстрее, чем FastKAN.

В случае с одномерным входным вектором RBF-нейронная сеть и FastKAN равны и по преобразованию, и по скорости.

То есть в итоге мы имеем более быструю и потенциально более точную модель.

4.5 Проклятье и благословение размерности

При обсуждении проклятия размерности часто игнорируют благословение размерности, считая его учтённым по умолчанию, что ошибочно. Эти явления являются двумя сторонами одной медали: увеличение размерности может как приносить преимущества, так и создавать недостатки в зависимости от контекста и подхода к решению задачи.

Проклятие размерности – термин, описывающий различные проблемы, которые возникают при анализе и обработке данных в многомерных пространствах, но не проявляются в пространствах с малым количеством измерений, например в привычном трёхмерном пространстве. Этот термин был введён Ричардом Беллманом в 1957 году.

Благословение размерности – термин, который, вопреки ожиданиям о проклятии размерности, указывает на то, что в высокоразмерных пространствах могут возникать и положительные эффекты, позволяющие решать задачи проще, чем ожидалось. Он был введён в конце 1990-х годов Дэвидом Донохо.

Суть проклятия размерности в том, что при увеличении числа измерений объём пространства растёт экспоненциально быстро, а имеющиеся данные становятся «разреженными», то есть их оказывается слишком мало для надёжного анализа. Это особенно важно, когда количество данных невелико относительно внутренней размерности самих данных. При этом также может экспоненциально расти сложность модели при увеличении количества измерений входных данных.

Также часто пространство начинает демонстрировать явление концентрации меры: величины, зависящие от случайных переменных, в высоких размерностях концентрируются вокруг своих средних значений, и вероятность больших отклонений стремится к нулю. К примеру, так ведут себя нормы векторов, скалярные произведения, углы между векторами, эвклидовы расстояния и т. п.

Однако все эти выводы в основном опираются на синтетические сценарии. На практике же нередко преобладает благословение размерности.

Суть в том, что машинное обучение связано с отнесением экземпляров к их источникам, а классовые метки символизируют разные процессы. Аргумент о проклятии исходит из предположения, что все данные приходят из одного процесса. Если бы это было так, машинное обучение теряло бы смысл. Однако, если учесть несколько процессов, потеря контраста может стать полезной, так как помогает находить ближайший и наиболее связанный экземпляр, делая расстояния более значимыми в высоких размерностях. На практике вектора часто подвержены одинаковым внутриклассовым корреляциям между признаками внутри одного и того же класса. Это приводит к тому, что эффективная размерность одного класса оказывается ниже, чем всё пространство данных, а значит, позволяет моделям машинного обучения разделять классы более эффективно.

Теперь перейдём к MLP. Стереотип о том, что эти сети подвержены проклятию размерности, выглядит чрезмерно громким, так как они (как и многие другие модели машинного обучения) в одних условиях могут страдать от проклятия размерности, а в других – нет. Всё зависит от конкретных обстоятельств, в которых мы их рассматриваем. В контексте нейронных сетей, и в частности MLP, обычно говорят об экспоненциальном росте сложности самой модели при увеличении входного пространства признаков и определённой желаемой точности.

К примеру, есть исследование Эндрю Р. Баррона «Оценки универсальной аппроксимации для суперпозиций сигмоидной функции», в котором анализируется MLP-архитектура, предложенная Цыбенко. Баррон доказывает, что при сходимости данного интеграла (нормы Баррона):

$C_f = \int_{\mathbb{R}^d} |\omega| |\hat{f}(\omega)| \, d\omega<\infty$

где $\hat{f}$ – это преобразование Фурье целевой функцииколичество нейронов, необходимое для достижения заданной точности аппроксимации, растёт полиномиально, а не экспоненциально с увеличением размерности входных данных. То есть, если преобразование Фурье целевой функции не содержит больших высокочастотных элементов и убывает достаточно быстро, то проклятие размерности можно считать разрушенным.

Всё аналогично и для B-spline KAT: предложенная модель также разрушает проклятие размерности, но лишь при соблюдении ряда условий. А именно, если целевую функциюможно разложить на блоки $\Phi,$ обладающие непрерывными производными порядка k+1, и при этом константав ошибке аппроксимации не растёт экспоненциально с увеличением размерности входных данных, то тогда проклятие размерности можно считать разрушенным.

Кстати, на MathOverflow есть обсуждение, в котором ставят под сомнение точность утверждения о разрушении проклятия размерности и просят уточнить условия.

В целом же это одна из проблем множества статей, рассматривающих B-spline KAN, поскольку некорректно утверждать, что MLP якобы подвержен проклятию размерности, а KAN – нет. На деле всё определяется конкретными предположениями и условиями применения той или иной модели.

4.6 Итог и обобщение информации с помощью множеств

В этой статье одной из целей, которую я ставил, было показать, что многие известные архитектуры сетей прямого распространения можно объединить и рассматривать через призму KAN, даже те, которые с первого взгляда не являются ими. Это дает возможность лучше понять, в чем они схожи и чем отличаются.

Также предлагаю теперь посмотреть как то, что мы обсуждали, будет выглядеть в контексте множеств. Однако сразу предупреждаю, что данная визуализация не охватывает все варианты и подмножества которые мы обсуждали, а для удобства восприятия включены только основные. В качестве примера пересечения множеств можно рассматривать комбинацию двух вариантов.

Список еще некоторых вариантов KAN можно найти в данном репозитории на GitHub.

И напоследок хотел сказать, что в планах так же написаь статью, где я покажу 2 новых варианта KAN. На данный момент в интернете подобные реализации мне еще не встречались.

Так что будет интересно!

Источники

Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems. Сybenko, G. (1989)

Lower bounds for approximation by MLP neural networks. Maiorov, Vitaly; Pinkus, Allan (April 1999)

Multilayer feedforward networks are universal approximators. Hornik, Kurt; Stinchcombe, Maxwell; White, Halbert (January 1989)

Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Leshno, Moshe; Lin, Vladimir Ya.; Pinkus, Allan; Schocken, Shimon (March 1992)

Support-Vector Networks. Cortes, Corinna; Vapnik, Vladimir (1995)

Mercer's theorem. J. Mercer (May 1909)

Introduction to Machine Learning: Class Notes 67577. Shashua, Amnon (2009)

Universal Approximation Using Radial-Basis-Function Networks. Park, J.; I. W. Sandberg (Summer 1991)

KAN: Kolmogorov-Arnold Networks. Ziming Liu et al

Kolmogorov-Arnold Networks are Radial Basis Function Networks. Ziyao Li

When is "Nearest Neighbor" Meaningful? Beyer, K.; Goldstein, J.; Ramakrishnan, R.; Shaft, U (1999)

A survey on unsupervised outlier detection in high-dimensional numerical data. Zimek, A.; Schubert, E.; Kriegel, H.P. (2012)

Shell Theory: A Statistical Model of Reality. Lin, Wen-Yan; Liu, Siying; Ren, Changhao; Cheung, Ngai-Man; Li, Hongdong; Matsushita, Yasuyuki (2021)

Utilizing Geometric Anomalies of High Dimension: When Complexity Makes Computation Easier. Kainen, Paul C. (1997)

Blessing of dimensionality: mathematical foundations of the statistical physics of data. Gorban, Alexander N.; Tyukin, Ivan Y. (2018)

Universal Approximation Bounds for Superpositions of a Sigmoidal Function.
Andrew R. Barron

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5 янв. 2025 г.

22.01.26 07:48 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
22.01.26 07:50 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
22.01.26 10:42 Tonerdomark

I got my money back from the Elon Musk scam. It cost me over 1 BTC and $55,000 in Dogecoin. Scammers vowed to double investments. Their sites seemed real. Fraud was tough to catch early. They hooked me with fast doubles in weeks or months. Videos showed Musk promising giveaways and gains. I bought in. I sent Bitcoin and Dogecoin in bits at first. Small sends worked. Then I wired my full savings. It vanished quick. No answers came. Bank account empty. Bills piled up. Loans covered rent and food. Sleep fled. Stress hit hard. Life crumbled. A friend spotted my trouble. He told of his scam loss last year. Same old plays. He pointed me to Sylvester Bryant, a recovery expert. Email Yt7cracker@gmail. com. WhatsApp +1 512 577 7957 or +44 7428 662701. Sylvester acted fast. He tracked blockchain trails. Dealt with exchanges. Outsmarted the scammers. In weeks, my Bitcoin came back. Even their phony profits too. Debts gone. Life back on track. Got hit? Contact him now.
22.01.26 19:25 Angela_Moore

Help to recover money from elon musk giveaway scam I got my money back from the Elon Musk scam. It cost me over 1 BTC and $55,000 in Dogecoin. Scammers vowed to double investments. Their sites seemed real. Fraud was tough to catch early. They hooked me with fast doubles in weeks or months. Videos showed Musk promising giveaways and gains. I bought in. I sent Bitcoin and Dogecoin in bits at first. Small sends worked. Then I wired my full savings. It vanished quick. No answers came. Bank account empty. Bills piled up. Loans covered rent and food. Sleep fled. Stress hit hard. Life crumbled. A friend spotted my trouble. He told of his scam loss last year. Same old plays. He pointed me to Sylvester Bryant, a recovery expert. Email Yt7cracker@gmail. com. WhatsApp +1 512 577 7957 or +44 7428 662701. Sylvester acted fast. He tracked blockchain trails. Dealt with exchanges. Outsmarted the scammers. In weeks, my Bitcoin came back. Even their phony profits too. Debts gone. Life back on track. Got hit? Contact him now.
23.01.26 07:35 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
23.01.26 07:35 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
26.01.26 10:36 alksnismareks

It all started when I decided to explore online trading as a way to grow my savings. Like many, I trusted what appeared to be a legitimate platform, only to find myself trapped in a nightmare. After making consistent trades and finally deciding to withdraw my profits, I was met with silence. My account was suddenly restricted—no warning, no explanation. Every attempt to contact the broker went unanswered or was met with vague, dismissive replies. For three long, agonizing months, I lived in uncertainty. I couldn’t sleep at night. I replayed every email, every transaction, wondering if I’d made a mistake. But deep down, I knew the truth: I hadn’t done anything wrong. The broker had simply decided to lock me out and keep my money. During that time, I felt completely powerless—like I was shouting into a void. The stress affected my health, my relationships, and my ability to focus on anything else. There were days I truly believed that $167,000 was gone forever, lost to the shadows of the unregulated online trading world. I even began to accept it as a painful lesson—one that would cost me dearly but might teach me to be more cautious in the future. But something inside me refused to surrender completely. That’s when I discovered TechY Force Cyber Retrieval. At first, I was cautious—after being scammed once, I didn’t want to fall victim again. But everything about TechY Force felt different. They were transparent from the start. No grand promises, no pressure tactics. Just clear, professional communication and a deep understanding of how these fraudulent brokers operate. Most importantly, they are a licensed specialist in binary options and forex fund recovery, which gave me the confidence to move forward. From our very first consultation, their team treated my case with urgency and empathy. They walked me through the entire process, explained the legal and technical avenues available, and assured me they would handle every detail. They collected documentation, analyzed transaction trails, and engaged directly with the payment processors and the broker using precise, strategic methods I never could have navigated on my own. What happened next was nothing short of miraculous. Within weeks, the broker—who had ignored me for months—began responding. And then, without any further drama or delays, my full $167,000 USD was returned to me. No deductions. No hidden fees. Just clean, complete recovery. The relief I felt was indescribable. It wasn’t just about the money—it was about reclaiming control, restoring trust, and proving that even in the face of deception, there are still good people who fight for what’s right. If you’ve been locked out of your trading account, scammed by a fake investment platform, or had your funds unjustly withheld, please know this: you are not alone, and your money may not be lost forever. Thanks to TechY Force Cyber Retrieval, I got my life back. Their expertise, integrity, and unwavering commitment turned my despair into deliverance. I cannot recommend them highly enough. To anyone reading this in distress: don’t give up. Reach out. Take that step. Because if someone like me—broken, doubtful, and nearly hopeless—can recover every dollar… so can you. WhatsApp them + 156 172 63 697 With heartfelt thanks and renewed hope, — A Recovered and Grateful Client
26.01.26 23:21 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.01.26 23:21 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.01.26 23:21 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.01.26 01:18 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
27.01.26 01:19 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
27.01.26 09:29 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.01.26 09:29 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.01.26 09:32 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.01.26 05:03 joyo

The digital world of cryptocurrency offers big chances, but it also hides tricky scams. Losing your crypto to fraud feels awful. It can leave you feeling lost and violated. This guide tells you what to do right away if a crypto scam has hit you. These steps can help you get funds back or stop more trouble. Knowing what to do fast can change everything,reach marie ([email protected] and whatsapp:+1 7127594675)
30.01.26 08:23 joseph67t

It's a joy to write this review. Since I began working with Marie at the beginning of 2018, the service has been outstanding. Hackers stole my monies, and I was frightened about how I would get them back. I didn't know where to begin, consequently it was a nightmare for me. But once my friend told me about ([email protected] and whatsap:+1 7127594675), things became simple for me. I'm glad she was able to get my bitcoin back so I could start trading on Binance again!
31.01.26 00:55 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
31.01.26 00:55 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
02.02.26 18:52 Christopherbelle

Sylvester Bryant is a top crypto recovery agent! Then I contacted them with my story that i have been scammed. It took time, yet my stolen crypto was recovered . Need help? Reach out to Sylvester on WhatsApp at +1 512 577 7957 or +44 7428 662701. Or email yt7cracker@gmail . com.
03.02.26 08:05 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
03.02.26 08:05 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.02.26 16:23 borutaralf

GREAT WHIP RECOVERY CYBER SERVICES TRUSTED EXPERTS IN ONLINE RECOVERY SOLUTIONS PHONE CALL:+1(406)2729101 I was unfortunately deceived and scammed out of $88,000 by someone I trusted to manage my funds during a transaction we carried out together. The experience left me deeply disappointed and hurt, realizing that someone could betray that level of trust without any remorse. Determined to seek justice and recover what was stolen, I began searching for legal assistance and came across numerous testimonials about GREAT WHIP RECOVERY CYBER SERVICES, a group known for helping victims recover lost funds. From what I learned, they have successfully assisted many people facing similar situations, returning stolen funds to their rightful owners in a remarkably short time. In my case, the GREAT WHIP RECOVERY CYBER SERVICES were able to recover my funds within just 48 hours, which was truly unbelievable. Even more reassuring was the fact that the scammer was identified, located, and eventually arrested by local authorities in his region. That outcome brought a great sense of relief and closure. I hope this information helps others who have lost their hard-earned money due to misplaced trust. If you’re in a similar situation, you can contact them through their info below to seek help in recovering your stolen funds. Email: [email protected] Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site Phone Call:+1(406)2729101
04.02.26 16:24 borutaralf

GREAT WHIP RECOVERY CYBER SERVICES TRUSTED EXPERTS IN ONLINE RECOVERY SOLUTIONS PHONE CALL:+1(406)2729101 I was unfortunately deceived and scammed out of $88,000 by someone I trusted to manage my funds during a transaction we carried out together. The experience left me deeply disappointed and hurt, realizing that someone could betray that level of trust without any remorse. Determined to seek justice and recover what was stolen, I began searching for legal assistance and came across numerous testimonials about GREAT WHIP RECOVERY CYBER SERVICES, a group known for helping victims recover lost funds. From what I learned, they have successfully assisted many people facing similar situations, returning stolen funds to their rightful owners in a remarkably short time. In my case, the GREAT WHIP RECOVERY CYBER SERVICES were able to recover my funds within just 48 hours, which was truly unbelievable. Even more reassuring was the fact that the scammer was identified, located, and eventually arrested by local authorities in his region. That outcome brought a great sense of relief and closure. I hope this information helps others who have lost their hard-earned money due to misplaced trust. If you’re in a similar situation, you can contact them through their info below to seek help in recovering your stolen funds. Email: [email protected] Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site Phone Call:+1(406)2729101
04.02.26 17:11 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
05.02.26 12:07 Thomas Muller

YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK CALL:+1(406)2729101 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with, website https://greatwhiprecoveryc.wixsite.com/greatwhip-site text +1(406)2729101 email [email protected]
05.02.26 15:46 feliciabotezatu

Losing access to your cryptocurrency can be devastating—whether you’ve been scammed, hacked, or locked out due to a forgotten password. Many assume their digital assets are gone forever. But with the right expertise, recovery is not only possible—it’s our daily reality. At TECHY FORCE CYBER RETRIEVAL (TFCR), we’re a globally recognized, fully legitimate crypto recovery service dedicated to helping victims reclaim lost or stolen digital assets—safely, ethically, and effectively. Who We Are Backed by a team of certified blockchain forensic analysts, cybersecurity specialists, and ethical hackers, TFCR has recovered millions of dollars in Bitcoin, Ethereum, USDT, and other major cryptocurrencies for clients worldwide. We specialize in cases involving: - Investment scams and fake platforms - Wallet hacks and unauthorized transactions - Forgotten passwords, seed phrases, or corrupted backups - Inaccessible hardware or software wallets Our mission is clear: Help you recover what’s rightfully yours—with honesty, transparency, and proven results. How We Work 1. Confidential Case Review Share your situation with us—no cost, no obligation. We assess whether your case is recoverable based on transaction data, wallet details, and loss type. 2. Advanced Blockchain Forensics Using industry-leading tools, we trace your funds across blockchains, identify destination addresses, and determine if assets are held on exchanges or recoverable platforms—even after complex laundering attempts. 3. Custom Recovery Execution Depending on your case, we: - Reconstruct access to locked wallets using secure decryption methods - Engage with exchanges or payment processors to freeze or retrieve funds - Provide forensic reports to support legal or compliance actions - Negotiate with third parties when appropriate and safe 4. Secure Return & Prevention Advice Recovered assets go directly to a wallet you control. We also offer practical guidance to help you avoid future losses—because security starts after recovery. Why Choose TFCR? No Recovery, No Fee – You only pay upon successful retrieval Legitimate & Transparent – No upfront payments, no hidden costs Global Expertise – Proven success across 50+ countries Ethical Standards – All actions comply with cybersecurity and privacy best practices While crypto threats grow daily, so does our resolve. At TECHY FORCE CYBER RETRIEVAL, we don’t just track transactions—we restore trust, hope, and financial peace of mind. Don’t give up on your crypto. Act now—before critical evidence disappears. 📧 Email: [email protected] 🌐 Visit: Official https://techyforcecyberretrieval.com Website] 🕒 Available 24/7 for urgent cases Your crypto may be missing—but with TFCR, it’s never truly lost. ©️ 2026 TECHY FORCE CYBER RETRIEVAL — Trusted. Professional. Results-Driven.
05.02.26 15:52 harryjones5

How Can I Contact a Cryptocurrency Recovery Company? Visit iFORCE HACKER RECOVERY I realize how volatile and thrilling cryptocurrency can be. After joining a Telegram-based service, I made consistent profits for six months before unexpected faults deprived me of approximately $343,000. Withdrawal blunders, little help, and rising dread kept me stuck. I then discovered iForce Hacker Recovery from positive reviews. They replied swiftly, handled my issue professionally, and walked me through every step. My valuables were returned within a week, giving me back my confidence. I heartily recommend their dependable, professional aid services. Contact Info: Website address: htt p:// iforcehackers. co m. Email: iforcehk @ consultant .co m WhatsApp: +1 240 803-3706
06.02.26 14:44 feliciabotezatu

Losing access to your cryptocurrency can be devastating—whether you’ve been scammed, hacked, or locked out due to a forgotten password. Many assume their digital assets are gone forever. But with the right expertise, recovery is not only possible—it’s our daily reality. At TECHY FORCE CYBER RETRIEVAL (TFCR), we’re a globally recognized, fully legitimate crypto recovery service dedicated to helping victims reclaim lost or stolen digital assets—safely, ethically, and effectively. Who We Are Backed by a team of certified blockchain forensic analysts, cybersecurity specialists, and ethical hackers, TFCR has recovered millions of dollars in Bitcoin, Ethereum, USDT, and other major cryptocurrencies for clients worldwide. We specialize in cases involving: - Investment scams and fake platforms - Wallet hacks and unauthorized transactions - Forgotten passwords, seed phrases, or corrupted backups - Inaccessible hardware or software wallets Our mission is clear: Help you recover what’s rightfully yours—with honesty, transparency, and proven results. How We Work 1. Confidential Case Review Share your situation with us—no cost, no obligation. We assess whether your case is recoverable based on transaction data, wallet details, and loss type. 2. Advanced Blockchain Forensics Using industry-leading tools, we trace your funds across blockchains, identify destination addresses, and determine if assets are held on exchanges or recoverable platforms—even after complex laundering attempts. 3. Custom Recovery Execution Depending on your case, we: - Reconstruct access to locked wallets using secure decryption methods - Engage with exchanges or payment processors to freeze or retrieve funds - Provide forensic reports to support legal or compliance actions - Negotiate with third parties when appropriate and safe 4. Secure Return & Prevention Advice Recovered assets go directly to a wallet you control. We also offer practical guidance to help you avoid future losses—because security starts after recovery. Why Choose TFCR? No Recovery, No Fee – You only pay upon successful retrieval Legitimate & Transparent – No upfront payments, no hidden costs Global Expertise – Proven success across 50+ countries Ethical Standards – All actions comply with cybersecurity and privacy best practices While crypto threats grow daily, so does our resolve. At TECHY FORCE CYBER RETRIEVAL, we don’t just track transactions—we restore trust, hope, and financial peace of mind. Don’t give up on your crypto. Act now—before critical evidence disappears. Email: [email protected] Visit: Official https://techyforcecyberretrieval.com Website] Available 24/7 for urgent cases Your crypto may be missing—but with TFCR, it’s never truly lost. ©️ 2026 TECHY FORCE CYBER RETRIEVAL — Trusted. Professional. Results-Driven.
07.02.26 00:44 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
07.02.26 00:44 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
07.02.26 04:43 Matt Kegan

Reach out to SolidBlock Forensics if you want to get back your coins from fake crypto investment or your wallet was compromised and all your coins gone. SolidBlock Forensics provide deep ethical analysis and investigation that enables them to trace these schemes, and recover all your funds. Their services are professional and reliable. http://www.solidblockforensics.com
07.02.26 17:31 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
10.02.26 23:52 frankqq

It is a pleasure to write this review. Since I began working with Marie in early 2018, the service has been outstanding. My coins were stolen by hackers, and I was afraid I wouldn't be able to recover them. It was a nightmare for me because I didn't know where to start. But after my friend told me about [email protected] and whatsapp:+1 7127594675, things became simple for me. I'm glad she was able to get my bitcoin back so I could start trading again.
11.02.26 05:50 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.02.26 05:50 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.02.26 23:55 brouwerspatrick8

I’ve always believed that sustainability begins at home—not just in how we recycle or conserve energy, but in the very structures we live in. For years, I dreamed of building a zero-waste neighborhood where every house functions like a living ecosystem: solar-powered, water-wise, and crowned with rooftop greenhouses that feed families and filter air. It wasn’t just architecture—it was my vision for a quieter, cleaner future. To make it real, I turned to Bitcoin. Not as a speculative bet, but as a long-term store of value aligned with my values—decentralized, transparent, and independent of broken systems. Over seven years, I poured savings, side income, and relentless discipline into building a $680,000 crypto portfolio. Every coin had a purpose: permits, materials, and community partnerships. My dream had a balance sheet. Then, in one exhausted, distracted moment, it all collapsed. It was November 2025. I was juggling contractor delays, city inspections, and endless design revisions. My nerves were frayed, my coffee pot never empty. When a “Ledger Live Update” notification popped up, I didn’t think twice. The interface looked identical—same logo, same layout. I entered my credentials… and within seconds, the app disappeared. My wallet balance dropped to zero. I sat frozen. My stomach dropped. All that work—years of sacrifice—gone in a blink. The days that followed were dark. I scoured forums, filed reports, and replayed my mistake on loop. Guilt ate at me. How could I have been so careless? My greenhouse renderings sat untouched. My dream felt like a cruel joke. Just when I was ready to walk away, I stumbled upon a newsletter about green innovation. Tucked between articles on carbon-neutral cities and next-gen solar panels was a short feature on *Digital Light Solution*—a specialized team that helps victims of crypto theft recover stolen assets. Skeptical but desperate, I reached out. What followed wasn’t magic—but it was close to it. Their team treated my case with urgency and compassion. They traced the transaction trail, identified the laundering path, and worked with exchanges to freeze what they could. Within weeks, they’d recovered a significant portion of my funds—enough to restart. Today, I’m not just rebuilding my portfolio—I’m breaking ground on my prototype greenhouse. And every beam, every pane of glass, carries the lesson I learned: that even in our most vulnerable moments, there’s still light to be found. [email protected] Telegram ——digitallightsolution website https://digitallightsolution.com/ WHAT'S APP https://wa.link/989vlf
12.02.26 23:56 brouwerspatrick8

I’ve always believed that sustainability begins at home—not just in how we recycle or conserve energy, but in the very structures we live in. For years, I dreamed of building a zero-waste neighborhood where every house functions like a living ecosystem: solar-powered, water-wise, and crowned with rooftop greenhouses that feed families and filter air. It wasn’t just architecture—it was my vision for a quieter, cleaner future. To make it real, I turned to Bitcoin. Not as a speculative bet, but as a long-term store of value aligned with my values—decentralized, transparent, and independent of broken systems. Over seven years, I poured savings, side income, and relentless discipline into building a $680,000 crypto portfolio. Every coin had a purpose: permits, materials, and community partnerships. My dream had a balance sheet. Then, in one exhausted, distracted moment, it all collapsed. It was November 2025. I was juggling contractor delays, city inspections, and endless design revisions. My nerves were frayed, my coffee pot never empty. When a “Ledger Live Update” notification popped up, I didn’t think twice. The interface looked identical—same logo, same layout. I entered my credentials… and within seconds, the app disappeared. My wallet balance dropped to zero. I sat frozen. My stomach dropped. All that work—years of sacrifice—gone in a blink. The days that followed were dark. I scoured forums, filed reports, and replayed my mistake on loop. Guilt ate at me. How could I have been so careless? My greenhouse renderings sat untouched. My dream felt like a cruel joke. Just when I was ready to walk away, I stumbled upon a newsletter about green innovation. Tucked between articles on carbon-neutral cities and next-gen solar panels was a short feature on *Digital Light Solution*—a specialized team that helps victims of crypto theft recover stolen assets. Skeptical but desperate, I reached out. What followed wasn’t magic—but it was close to it. Their team treated my case with urgency and compassion. They traced the transaction trail, identified the laundering path, and worked with exchanges to freeze what they could. Within weeks, they’d recovered a significant portion of my funds—enough to restart. Today, I’m not just rebuilding my portfolio—I’m breaking ground on my prototype greenhouse. And every beam, every pane of glass, carries the lesson I learned: that even in our most vulnerable moments, there’s still light to be found. [email protected] Telegram ——digitallightsolution website https://digitallightsolution.com/ WHAT'S APP https://wa.link/989vlf
13.02.26 00:17 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.02.26 00:17 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.02.26 02:16 Ralf Boruta

GREAT WHIP RECOVERY CYBER SERVICES TRUSTED EXPERTS IN ONLINE RECOVERY SOLUTIONS PHONE CALL:+1(406)2729101 I was unfortunately deceived and scammed out of $88,000 by someone I trusted to manage my funds during a transaction we carried out together. The experience left me deeply disappointed and hurt, realizing that someone could betray that level of trust without any remorse. Determined to seek justice and recover what was stolen, I began searching for legal assistance and came across numerous testimonials about GREAT WHIP RECOVERY CYBER SERVICES, a group known for helping victims recover lost funds. From what I learned, they have successfully assisted many people facing similar situations, returning stolen funds to their rightful owners in a remarkably short time. In my case, the GREAT WHIP RECOVERY CYBER SERVICES were able to recover my funds within just 48 hours, which was truly unbelievable. Even more reassuring was the fact that the scammer was identified, located, and eventually arrested by local authorities in his region. That outcome brought a great sense of relief and closure. I hope this information helps others who have lost their hard-earned money due to misplaced trust. If you’re in a similar situation, you can contact them through their info below to seek help in recovering your stolen funds. Email: [email protected] Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site Phone Call:+1(406)2729101
13.02.26 02:16 Ralf Boruta

GREAT WHIP RECOVERY CYBER SERVICES TRUSTED EXPERTS IN ONLINE RECOVERY SOLUTIONS PHONE CALL:+1(406)2729101 I was unfortunately deceived and scammed out of $88,000 by someone I trusted to manage my funds during a transaction we carried out together. The experience left me deeply disappointed and hurt, realizing that someone could betray that level of trust without any remorse. Determined to seek justice and recover what was stolen, I began searching for legal assistance and came across numerous testimonials about GREAT WHIP RECOVERY CYBER SERVICES, a group known for helping victims recover lost funds. From what I learned, they have successfully assisted many people facing similar situations, returning stolen funds to their rightful owners in a remarkably short time. In my case, the GREAT WHIP RECOVERY CYBER SERVICES were able to recover my funds within just 48 hours, which was truly unbelievable. Even more reassuring was the fact that the scammer was identified, located, and eventually arrested by local authorities in his region. That outcome brought a great sense of relief and closure. I hope this information helps others who have lost their hard-earned money due to misplaced trust. If you’re in a similar situation, you can contact them through their info below to seek help in recovering your stolen funds. Email: [email protected] Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site Phone Call:+1(406)2729101
13.02.26 18:29 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.02.26 18:29 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.02.26 23:59 Lilyfox

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18.02.26 00:01 Lilyfox

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18.02.26 03:23 walterlindahi9

This past January, my world came crashing down. I lost nearly $42,000 of my hard-earned savings to a sophisticated Solana-based crypto scam. At first, it all seemed legitimate: sleek website, professional whitepaper, even glowing testimonials from “investors.” I’d done my homework, or so I thought. The promise of high returns in a volatile market felt like my ticket to financial freedom. For the first few months, everything appeared to be working. My portfolio showed steady gains. I remember checking my wallet balance daily, feeling a mix of pride and relief. I’ve cracked the code to building real wealth. Then, without warning, the platform vanished. Wallet addresses went dead. Support channels disappeared, and my funds were gone in an instant. The emotional fallout was worse than the financial loss. Sleepless nights became the norm. Anxiety gnawed at me constantly. I replayed every decision in my head, blaming myself for being naive. I vowed never to trust anyone again, not influencers, not experts, not even my own judgment. But giving up wasn’t an option. I owed it to myself and to my future to fight back. So I began digging. I scoured Reddit threads, filed reports with blockchain analytics firms, and even contacted local authorities (though they offered little help). The more I searched, the more overwhelmed I became, lost in a labyrinth of technical jargon, dead ends, and predatory recovery services asking for upfront fees. Then, through a survivor’s forum, I stumbled upon TechY Force Cyber Retrieval. Skeptical but desperate, I reached out. What set them apart wasn’t just their expertise; it was their empathy. They didn’t make wild promises. Instead, they walked me through how crypto tracing works, what success looks like, and what realistic timelines are. No pressure. No false hope. Within weeks, their forensic team identified transaction trails linked to the scam wallet. Using on-chain analysis and coordination with exchanges, they flagged suspicious activity and initiated recovery protocols. It wasn’t magic, but it was methodical, transparent, and grounded in real blockchain intelligence. Today, I’m cautiously optimistic. While not all funds have been recovered yet, TechY Force has already secured a significant portion and, more importantly, restored my sense of agency. I’m sleeping again. I’m healing. If you’ve been scammed, know this: you’re not alone, and you’re not foolish. Crypto fraud preys on hope, but that same hope can fuel your comeback. Don’t suffer in silence. Reach out. Ask questions. And never let a scammer steal your future along with your funds. WhatsApp +1(561) 726 3697 Mail. Techyforcecyberretrieval(@)consultant(.)com Telegram (@)TechCyberforc
22.02.26 03:48 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.02.26 03:49 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.02.26 18:58 Natasha Williams

I am Natasha Williams from Dallas. I want to share my testimony to encourage anyone who has ever fallen victim to a scam or fraud. Some time ago, I was defrauded by some fraudulent cryptocurrency investment organization online, I was a victim and I lost a huge amount of money, $382,000. I felt angry, disappointed and helpless but I refused to give up and stay calm. I came across this agency, GREAT WHIP RECOVERY CYBER SERVICES.. who helped people recover their money from scammers and the testimonies I saw were quite amazing. And I decided to contact them. I gathered every piece of evidence, chats, receipts, account details, and messages and reported the case to the agency, GREAT WHIP RECOVERY CYBER SERVICES. After 73hours of follow up and not losing faith, the fraudster was traced and held accountable and I recovered all my money back. I highly recommend, GREAT WHIP RECOVERY CYBER SERVICES agency if you have ever fallen victim to scammers, you can contact them. Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site email: [email protected] Call Line: +1(406)2729101
22.02.26 19:00 Natasha Williams

I am Natasha Williams from Dallas. I want to share my testimony to encourage anyone who has ever fallen victim to a scam or fraud. Some time ago, I was defrauded by some fraudulent cryptocurrency investment organization online, I was a victim and I lost a huge amount of money, $382,000. I felt angry, disappointed and helpless but I refused to give up and stay calm. I came across this agency, GREAT WHIP RECOVERY CYBER SERVICES.. who helped people recover their money from scammers and the testimonies I saw were quite amazing. And I decided to contact them. I gathered every piece of evidence, chats, receipts, account details, and messages and reported the case to the agency, GREAT WHIP RECOVERY CYBER SERVICES. After 73hours of follow up and not losing faith, the fraudster was traced and held accountable and I recovered all my money back. I highly recommend, GREAT WHIP RECOVERY CYBER SERVICES agency if you have ever fallen victim to scammers, you can contact them. Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site email: [email protected] Call Line: +1(406)2729101
23.02.26 23:26 chongfook

As cryptocurrencies continue to reshape global finance in 2026, the risks have never been higher. From sophisticated phishing campaigns to fake wallet apps and investment scams, millions of investors face the devastating reality of lost or stolen digital assets. When your crypto vanishes, panic sets in—and that's when fraudsters strike again, posing as "recovery experts" to exploit your vulnerability. CONTACTS US Techyforcecyberretrieval(@)consultant(.)com https(://)techyforcecyberretrieval(.)com But there's a legitimate path forward. TECHY FORCE CYBER RETRIEVAL stands as the industry's most trusted crypto recovery company, combining advanced blockchain forensics, global partnerships, and a client-centric approach to help victims reclaim what was stolen. --- Why Recovery Is Possible—With the Right Team Cryptocurrency's decentralized, pseudonymous nature makes asset recovery complex—but not impossible. The blockchain is transparent. Every transaction leaves a trail. The challenge isn't finding the funds—it's having the expertise to follow that trail through mixers, bridges, and exchange deposits before they disappear forever. That's where TECHY FORCE CYBER RETRIEVAL excels. --- Our Proven Recovery Framework We don't believe in shortcuts, false promises, or upfront fees. Our process is built on transparency, forensic precision, and real results. Here's how we work: 1. Case Intake & Initial Assessment You begin by submitting a detailed report: compromised wallet addresses, transaction IDs, timestamps, and any communication with scammers. Our intake team reviews your case within hours to determine immediate next steps. 2. Blockchain Forensic Analysis Our specialists deploy proprietary tracking tools to map the movement of your stolen assets across multiple blockchains. We identify laundering patterns, exchange deposit addresses, and potential freezing points—building a clear investigative roadmap. 3. Global Partner Coordination Through established relationships with regulated exchanges, DeFi protocols, and compliance teams worldwide, we initiate direct communication to flag suspicious transactions and request asset freezes where legally permissible. 4. Legal & Regulatory Engagement When necessary, we collaborate with legal partners and law enforcement agencies to strengthen recovery efforts—especially in cases involving large-scale hacks or organized fraud rings. 5. Recovery Execution & Fund Return Once assets are secured, they're transferred directly to a new, secure wallet of your choice. We never hold your funds. And critically, we operate on a success-only model. You pay nothing unless we recover your assets. 6. Post-Recovery Security Guidance Recovery is only half the battle. We provide personalized recommendations to secure your remaining holdings—from hardware wallet setup to phishing awareness training—so you can move forward with confidence. --- What Sets TECHY FORCE CYBER RETRIEVAL Apart While countless "recovery services" flood the internet, few deliver legitimate results. Here's why we're consistently rated the best crypto recovery company in 2026: - Zero Upfront Fees – We only succeed when you do. No hidden charges. No bait-and-switch tactics. - Advanced Blockchain Intelligence – Our forensic tools track assets across Bitcoin, Ethereum, Solana, and 50+ other networks. - Global Reach – Partnerships with exchanges and regulatory bodies in North America, Europe, and Asia maximize recovery odds. - Client-First Communication – Weekly updates. Clear timelines. No ghosting. - Proven Track Record – Hundreds of successful recoveries in 2025–2026, with millions returned to rightful owners. --- Emerging Trends in 2026: What Victims Need to Know The threat landscape evolves constantly. This year's biggest risks include: - AI-Powered Phishing: Scammers now use deepfake voice and video to impersonate support staff. - Cross-Chain Bridge Exploits: Funds moved between networks are increasingly targeted. - Fake Recovery Services: Fraudsters pose as legitimate firms—always verify credentials before sharing information. TECHY FORCE CYBER RETRIEVAL stays ahead of these threats, continuously updating our tools and strategies to protect and serve our clients. CONTACTS US Techyforcecyberretrieval(@)consultant(.)com https(://)techyforcecyberretrieval(.)com --- Your Next Step If you've lost crypto to a scam, hack, or forgotten credentials, don't let despair—or another fraudster—steal your second chance. TECHY FORCE CYBER RETRIEVAL is accessible, transparent, and ready to help. Reach out today. Let our experts assess your case—and show you that even in 2026, stolen crypto doesn't have to stay lost forever. — TECHY FORCE CYBER RETRIEVAL Advanced Forensics. Global Reach. Your Recovery.
24.02.26 15:31 [email protected]`

Like many others, I was drawn in by the allure of cryptocurrency and the promise of financial freedom. When I encountered a self-proclaimed "crypto guru" online, his confidence and flashy lifestyle convinced me that he held the key to success. Eager to learn, I parted with $15,000 for his exclusive course, believing it would grant me access to an elite trading group and lucrative market insights. Initially, my excitement was palpable; I truly thought I was on the verge of a breakthrough. However, that enthusiasm quickly curdled into dread. Once inside the group, the dynamic shifted from education to aggressive exploitation. Instead of genuine mentorship, members were relentlessly upsold on fake trading signals that yielded nothing but losses. The pressure escalated when we were encouraged to invest in a supposed "private pool," which required an additional, staggering access fee of $60,000. It was only as I began to notice glaring inconsistencies and a complete lack of real results among the members that the fog lifted. I realized I hadn't joined a community of traders; I had walked into a sophisticated trap designed specifically to prey on newcomers like myself. The realization that the promises of wealth and insider knowledge were nothing more than a façade left me feeling vulnerable, deceived, and financially devastated. The dream of easy returns had turned into a heavy burden of regret. Desperate for a solution and refusing to let the fraudsters win, I began searching for help. That is when I discovered DIGITAL LIGHT SOLUTION, a firm specializing in online fraud investigations. Reaching out to them was the turning point. Their team approached my case with professionalism and empathy, immediately understanding the complexity of the scam. They guided me through the investigation process, uncovering the layers of deception used by the "guru" and his network. Thanks to their expertise and relentless pursuit of justice, I was able to navigate the aftermath of this ordeal with clarity rather than confusion. While the experience was a harsh lesson, connecting with DIGITAL LIGHT SOLUTION restored my hope and proved that there are still allies ready to fight against online exploitation. If you find yourself in a similar situation, do not lose hope—seek professional help immediately. Contact them directly Website https://digitallightsolution.com/ Email — Digitallightsolution(At)qualityservice(DOT)com What's App — https://wa.link/989vlf
24.02.26 15:32 [email protected]`

Like many others, I was drawn in by the allure of cryptocurrency and the promise of financial freedom. When I encountered a self-proclaimed "crypto guru" online, his confidence and flashy lifestyle convinced me that he held the key to success. Eager to learn, I parted with $15,000 for his exclusive course, believing it would grant me access to an elite trading group and lucrative market insights. Initially, my excitement was palpable; I truly thought I was on the verge of a breakthrough. However, that enthusiasm quickly curdled into dread. Once inside the group, the dynamic shifted from education to aggressive exploitation. Instead of genuine mentorship, members were relentlessly upsold on fake trading signals that yielded nothing but losses. The pressure escalated when we were encouraged to invest in a supposed "private pool," which required an additional, staggering access fee of $60,000. It was only as I began to notice glaring inconsistencies and a complete lack of real results among the members that the fog lifted. I realized I hadn't joined a community of traders; I had walked into a sophisticated trap designed specifically to prey on newcomers like myself. The realization that the promises of wealth and insider knowledge were nothing more than a façade left me feeling vulnerable, deceived, and financially devastated. The dream of easy returns had turned into a heavy burden of regret. Desperate for a solution and refusing to let the fraudsters win, I began searching for help. That is when I discovered DIGITAL LIGHT SOLUTION, a firm specializing in online fraud investigations. Reaching out to them was the turning point. Their team approached my case with professionalism and empathy, immediately understanding the complexity of the scam. They guided me through the investigation process, uncovering the layers of deception used by the "guru" and his network. Thanks to their expertise and relentless pursuit of justice, I was able to navigate the aftermath of this ordeal with clarity rather than confusion. While the experience was a harsh lesson, connecting with DIGITAL LIGHT SOLUTION restored my hope and proved that there are still allies ready to fight against online exploitation. If you find yourself in a similar situation, do not lose hope—seek professional help immediately. Contact them directly Website https://digitallightsolution.com/ Email — Digitallightsolution(At)qualityservice(DOT)com What's App — https://wa.link/989vlf
26.02.26 16:29 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.02.26 16:29 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.02.26 00:08 sanayoliver

I spend my days studying the mysteries of the universe, delving into black holes, quantum mechanics, and the nature of time itself. But apparently, the real black hole I should have been concerned about was my own memory. I encrypted my Bitcoin wallet to keep it as secure as possible. The problem? I promptly forgot the password. Classic, right? It didn't help that this wasn't just pocket change I was dealing with. No, I had $190,000 in Bitcoin sitting in that wallet, and my mind had decided to take a vacation, leaving me with absolutely no idea what that password was. The panic set in fast. My brain, which could solve some of the most complex physics equations, couldn't remember a 12-character password. It felt like my entire financial future was being sucked into a black hole, one I'd created myself. Desperate, I tried everything. I thought I could outsmart the system, using every trick I could think of. I tried variations of passwords I thought I might have used, analyzing them through the lens of my own behavioral patterns. I even resorted to good ol' brute force, typing random combinations for hours, hoping that maybe, just maybe, my subconscious would strike gold. Spoiler alert: it didn't. Each failed attempt made me feel more and more like a genius who'd locked themselves out of their own universe. In a final act of desperation, admitting that theoretical physics couldn't crack my own encryption, I contacted TechY Force Cyber Retrieval. From the moment I reached out, the difference was night and day. While I had been flailing in the dark, they approached my case with a precision that rivaled the calculations I do daily. They didn't promise miracles; they promised a methodical, advanced recovery process. Within a surprisingly short timeframe, they utilized specialized tools to bypass the mental block I couldn't overcome. When they finally recovered the wallet and confirmed the full $190,000 was intact and accessible, the relief was indescribable. It was as if I had pulled my financial future back from the event horizon just before it was lost forever. To anyone thinking they are too smart to lose their keys, or too logical to make such a mistake: don't wait until you are staring into the abyss. If you find yourself in a situation where your own memory has become your greatest enemy, trust the experts at TechY Force Cyber Retrieval. They turned my personal black hole into a success story, proving that sometimes, even the brightest minds need a little help to find the light. REACH OUT TO THEM ON MAIL [email protected]
27.02.26 02:04 elizabethmadison

My name is Elizabeth Madison currently living in New York. There was a time I felt completely broken. I had trusted a fraudulent bitcoin investment organization, who turned out to be a fraudster. I sent money, believing their sweet words and promises on the interest rate I will get back in return, only to realize later that I’ve been scammed. On the day of withdrawal there was no money in my account. The pain hit deep. I couldn’t sleep, I kept asking myself how I could have been so careless, meanwhile my mom was battling with a stroke and the expenses were too much. For days, I cried and blamed myself. The betrayal, the disappointment and my mom's health issues all of this stress made me want to give up on life. But one day, I decided that sitting in pain wouldn’t solve anything. I picked myself up and chose to fight for what I lost then I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how he helped people recover their funds from online fraud. I emailed all the transactions and paperwork I had with the fraudulent organization and they helped me recover all my lost money in just five days. If you have ever fallen victim to scammers, contact GREAT WHIP RECOVERY CYBER SERVICES to help you recover every penny you have lost. (Text +1(406)2729101) (Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site) (Email [email protected])
27.02.26 02:04 elizabethmadison

My name is Elizabeth Madison currently living in New York. There was a time I felt completely broken. I had trusted a fraudulent bitcoin investment organization, who turned out to be a fraudster. I sent money, believing their sweet words and promises on the interest rate I will get back in return, only to realize later that I’ve been scammed. On the day of withdrawal there was no money in my account. The pain hit deep. I couldn’t sleep, I kept asking myself how I could have been so careless, meanwhile my mom was battling with a stroke and the expenses were too much. For days, I cried and blamed myself. The betrayal, the disappointment and my mom's health issues all of this stress made me want to give up on life. But one day, I decided that sitting in pain wouldn’t solve anything. I picked myself up and chose to fight for what I lost then I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how he helped people recover their funds from online fraud. I emailed all the transactions and paperwork I had with the fraudulent organization and they helped me recover all my lost money in just five days. If you have ever fallen victim to scammers, contact GREAT WHIP RECOVERY CYBER SERVICES to help you recover every penny you have lost. (Text +1(406)2729101) (Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site) (Email [email protected])
27.02.26 15:57 luciajessy3

Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…
27.02.26 15:59 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.02.26 15:59 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.02.26 16:00 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.02.26 16:01 luciajessy3

Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…
27.02.26 16:01 luciajessy3

Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…
27.02.26 16:01 luciajessy3

Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…
01.03.26 10:48 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.03.26 10:48 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
03.03.26 14:09 Thomas Muller

YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK CALL:+1(406)2729101 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with, website https://greatwhiprecoveryc.wixsite.com/greatwhip-site text +1(406)2729101 email [email protected]
03.03.26 14:09 Thomas Muller

YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK CALL:+1(406)2729101 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with, website https://greatwhiprecoveryc.wixsite.com/greatwhip-site text +1(406)2729101 email [email protected]
04.03.26 07:21 Jane4

I lost about $600k Bitcoin last year, I searched around and tried to work with some recovery firm unfortunately I was scammed as well. This happened for months until I came across [email protected] They came to my rescue and all my funds were recovered within few days I'm so happy right now .
04.03.26 07:22 Jane4

I lost about $600k Bitcoin last year, I searched around and tried to work with some recovery firm unfortunately I was scammed as well. This happened for months until I came across [email protected] They came to my rescue and all my funds were recovered within few days I'm so happy right now .
04.03.26 12:25 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.03.26 12:25 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
06.03.26 13:36 CARL9090

In January, my life shifted in a way I never expected. I clicked a trading link given to me by someone I found on Telegram, believing it was legitimate. It looked professional. It felt secure. I trusted it. Until I tried to withdraw my money. Within seconds, everything was gone, transferred into a wallet claiming account without a trace. That was the moment the truth hit me: I had been scammed. The emotional fallout was brutal. For weeks, I couldn’t even speak about it. I thought people would judge me. I thought they’d say I should have known better. Then someone stepped in who changed everything Agent Jasmine Lopez ,She listened without judgment. She treated my fear as real and valid. She traced patterns, uncovered off-chain indicators, and identified wallet clusters linked to a larger scam network. She showed me that what happened wasn’t random it was organized and intentional. For the first time, I felt hope. Hearing that students, parents, and hardworking people had been targeted the same way made me realize this wasn’t stupidity. It was predation. We weren’t careless we were deliberately targeted and manipulated I’m still healing. The experience changed me. But it also reminded me that even in your darkest moment, there can be someone willing to shine a light. Contact her at [email protected] WHATSAPP +44 7478077894
07.03.26 07:46 Jane4

I lost about $600k Bitcoin last year, I searched around and tried to work with some recovery firm unfortunately I was scammed as well. This happened for months until I came across [email protected] They came to my rescue and all my funds were recovered within few days I'm so happy right now .
07.03.26 07:46 Jane4

I lost about $600k Bitcoin last year, I searched around and tried to work with some recovery firm unfortunately I was scammed as well. This happened for months until I came across [email protected] They came to my rescue and all my funds were recovered within few days I'm so happy right now .
07.03.26 08:39 Jane4

I lost about $600k Bitcoin last year, I searched around and tried to work with some recovery firm unfortunately I was scammed as well. This happened for months until I came across [email protected] They came to my rescue and all my funds were recovered within few days I'm so happy right now .
07.03.26 08:55 Jane4

I lost about $600k Bitcoin last year, I searched around and tried to work with some recovery firm unfortunately I was scammed as well. This happened for months until I came across [email protected] They came to my rescue and all my funds were recovered within few days I'm so happy right now .
07.03.26 09:40 Alena76

Most people have been scammed severally and they give up on their funds I'm saying these because I was a victim too After loosing 745,000 USD I lose my mind until I read about COIN HACK RECOVERY I decided to contact the company on: [email protected] and I'm glad I made the decision not to give up. they helped me to recover all my lost funds within two days.
07.03.26 10:37 Alena76

Most people have been scammed severally and they give up on their funds I'm saying these because I was a victim too After loosing 745,000 USD I lose my mind until I read about COIN HACK RECOVERY I decided to contact the company on: [email protected] and I'm glad I made the decision not to give up. they helped me to recover all my lost funds within two days.
07.03.26 10:37 Alena76

Most people have been scammed severally and they give up on their funds I'm saying these because I was a victim too After loosing 745,000 USD I lose my mind until I read about COIN HACK RECOVERY I decided to contact the company on: [email protected] and I'm glad I made the decision not to give up. they helped me to recover all my lost funds within two days.
07.03.26 17:49 Natasha Williams

I am Natasha Williams from Dallas. I want to share my testimony to encourage anyone who has ever fallen victim to a scam or fraud. Some time ago, I was defrauded by some fraudulent cryptocurrency investment organization online, I was a victim and I lost a huge amount of money, $382,000. I felt angry, disappointed and helpless but I refused to give up and stay calm. I came across this agency, GREAT WHIP RECOVERY CYBER SERVICES.. who helped people recover their money from scammers and the testimonies I saw were quite amazing. And I decided to contact them. I gathered every piece of evidence, chats, receipts, account details, and messages and reported the case to the agency, GREAT WHIP RECOVERY CYBER SERVICES. After 73hours of follow up and not losing faith, the fraudster was traced and held accountable and I recovered all my money back. I highly recommend, GREAT WHIP RECOVERY CYBER SERVICES agency if you have ever fallen victim to scammers, you can contact them. Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site email: [email protected] Call Line: +1(406)2729101
07.03.26 20:10 ericbank61

I never thought I’d be the one writing one of these stories. You hear about crypto scams, hacks, and lost fortunes, and you think, “That’s for other people. The careless ones.” I was careful. Or so I believed. It started with a sophisticated phishing attack. An email that looked identical to a legitimate exchange notification, a link to “verify my wallet security,” and a moment of distracted panic. I clicked. Within hours, my life savings in Bitcoin—a sum I’d been accumulating for five years—vanished from my private wallet. The transaction hash was a cold, unfeeling tombstone on the blockchain. My stomach dropped into a void. I felt physically ill. The police filed a report, but their knowledge ended at the edge of traditional finance. The exchange offered sympathy but no solutions. I was adrift, utterly hopeless. After weeks of despair, scouring forums in the dead of night, I found a thread mentioning Mighty Hacker Recovery. The name sounded almost too bold, like something from a cheesy movie. But the testimonials were detailed, sober, and from people who sounded just like me: desperate, betrayed, and out of options. With nothing left to lose, I reached out. Their intake process was professional but guarded. They asked for transaction IDs, wallet addresses, and a detailed timeline—no promises, just facts. A consultant named Leo became my point of contact. He had a calm, analytical voice that cut through my panic. “We don’t hack *into* systems,” he explained. “We follow the digital trail. We analyze the attack vector, trace the flow of funds through the blockchain’s transparency, and identify the weak points in the scammer’s own security. Sometimes, it’s about speed and outmaneuvering them before they can launder the assets.” What followed was a tense, silent partnership. I provided every shred of information I had, while Leo’s team worked in the shadows. There were days of silence that felt like years. Then, an update: they’d traced my BTC to a mixing service, a tool scammers use to obfuscate the trail. Mighty Hacker Recovery used advanced blockchain forensic techniques to peel back those layers. They discovered the scammer had made a critical error—a small portion of the funds was sent to a KYC-compliant exchange wallet. That was the chink in the armor. Using the immutable evidence from the blockchain and legal pressure channels they’d established with certain international platforms, they initiated a recovery claim. The process was complex, involving digital affidavits and proof of illicit origin. Three weeks after my first desperate email, Leo called. “We’ve secured a freeze on the destination wallet. The exchange is cooperating. We’re initiating the reversal.” I didn’t dare believe it until I saw it. Two days later, my wallet balance updated. My Bitcoin, minus Mighty Hacker Recovery’s contingency fee, was back. The relief wasn’t euphoric; it was a deep, trembling exhaustion, like waking up from a nightmare. They didn’t perform magic. They applied intense expertise, relentless persistence, and an intricate understanding of both the blockchain’s weaknesses and a scammer’s psychology. They gave me back more than my crypto; they gave me back a sense of agency in a landscape designed to make victims feel powerless. If you’re reading this from your own private hell of loss, know this: the trail never truly disappears. You just need the right team to follow it. For me, that was Mighty Hacker Recovery.
07.03.26 22:44 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
07.03.26 22:44 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.03.26 19:43 Michael Jensen

With the help and expertise of CapitalNode Analytics, i was able to get back my digital tokens from a fake investment platform. They are swift, precise and transparent in their operations.
12.03.26 15:04 Mike Franz

I recently ran into a serious issue with my cryptocurrency account that left me unable to access my bitcoin wallet. After several failed login attempts and repeated blocks from the system, I began to worry that I might lose access to my $415,000 permanently. Determined to fix the problem, After spending hours reading a review of GREAT WHIP RECOVERY CYBER SERVICES, on how they successfully assisted countless individuals in similar situations as mine. The process was stressful, but eventually the issue was resolved and I was able to regain access to my bitcoin wallet account. I’m immensely grateful to GREAT WHIP RECOVERY CYBER SERVICES for their incredible work, for those who need help, you can contact through the following channels: Phone Call: +1(406)2729101 Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Mail: [email protected]
12.03.26 15:05 Mike Franz

I recently ran into a serious issue with my cryptocurrency account that left me unable to access my bitcoin wallet. After several failed login attempts and repeated blocks from the system, I began to worry that I might lose access to my $415,000 permanently. Determined to fix the problem, After spending hours reading a review of GREAT WHIP RECOVERY CYBER SERVICES, on how they successfully assisted countless individuals in similar situations as mine. The process was stressful, but eventually the issue was resolved and I was able to regain access to my bitcoin wallet account. I’m immensely grateful to GREAT WHIP RECOVERY CYBER SERVICES for their incredible work, for those who need help, you can contact through the following channels: Phone Call: +1(406)2729101 Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Mail: [email protected]
15.03.26 20:22 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.03.26 20:22 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.03.26 20:22 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.03.26 12:01 [email protected]

I would like to highly recommend TOP RECOVERY EXPERT, the best in cryptocurrency recovery. I want the world to know how exceptional their services are. For years, I faced a very difficult time after being scammed out of $453,000 in Ethereum. It was devastating to realize that someone could steal from me without remorse after I trusted them. Determined to recover my funds legally, I began searching for reliable help and came across TOP RECOVERY EXPERT, the most professional recovery service I have ever found. With their expertise and support, I was able to recover my entire Ethereum wallet. I now understand that while many investment opportunities can seem too good to be true, professional guidance can make all the difference. Thanks to TOP RECOVERY EXPERT, I have regained not only my assets ETH but also my peace of mind and happiness. Their dedication and professionalism have truly changed my life. I am now the happiest person I have ever been, all because of their help. If you have been a victim of a crypto scam, I strongly advise you to reach out to TOP RECOVERY EXPERT. Contact Information: Text/Call: +1 (346) 980-9102 Email: [email protected] For more information visit his website: https://toprecoveryexpert2.wixsite.com/consultant
16.03.26 13:20 luciajessy3

There are many recommendations online, but not all of them are trustworthy. Unfortunately, some so-called “recovery services” are scams themselves and may try to take advantage of people who have already lost money. If you’ve been scammed, be extremely cautious about anyone promising guaranteed recovery — especially if they ask for upfront fees. Always do thorough research, verify credentials, and consider reporting the incident to. Cyberrefundexperts {A} Consultant {.} Com , Stay alert and protect yourself.
16.03.26 13:20 luciajessy3

There are many recommendations online, but not all of them are trustworthy. Unfortunately, some so-called “recovery services” are scams themselves and may try to take advantage of people who have already lost money. If you’ve been scammed, be extremely cautious about anyone promising guaranteed recovery — especially if they ask for upfront fees. Always do thorough research, verify credentials, and consider reporting the incident to. Cyberrefundexperts @ Consultant . Com , Stay alert and protect yourself.
18.03.26 15:27 keithwilson9899

ETHEREUM RECOVERY ASSISTANCE: CAPITAL CRYPTO RECOVER HELPED ME RECOVER $98,000 WORTH OF LOST ETH In cases of cryptocurrency scams, having accurate information and trusted support is essential. I would like to recommend Capital Crypto Recover Service, a professional team that specializes in assisting individuals with the recovery of lost or stolen Bitcoin and Ethereum (ETH). Their experienced experts are dedicated to helping victims of digital asset fraud by carefully analyzing each case, developing strategic recovery plans, Capital Crypto Recover Service knowledgeable team's primary goals are to satisfy clients and offer significant support and working diligently toward fund retrieval. The team is committed to providing reliable assistance and maintaining a high level of client satisfaction. Based on my assessment, their reputation professionalism and a strong commitment to their clients. If you have experienced a cryptocurrency loss, you can contacting them for further assistance Phone (Call/Text): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.03.26 15:27 keithwilson9899

ETHEREUM RECOVERY ASSISTANCE: CAPITAL CRYPTO RECOVER HELPED ME RECOVER $98,000 WORTH OF LOST ETH In cases of cryptocurrency scams, having accurate information and trusted support is essential. I would like to recommend Capital Crypto Recover Service, a professional team that specializes in assisting individuals with the recovery of lost or stolen Bitcoin and Ethereum (ETH). Their experienced experts are dedicated to helping victims of digital asset fraud by carefully analyzing each case, developing strategic recovery plans, Capital Crypto Recover Service knowledgeable team's primary goals are to satisfy clients and offer significant support and working diligently toward fund retrieval. The team is committed to providing reliable assistance and maintaining a high level of client satisfaction. Based on my assessment, their reputation professionalism and a strong commitment to their clients. If you have experienced a cryptocurrency loss, you can contacting them for further assistance Phone (Call/Text): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
08:03 Alena76

Most people have been scammed severally and they give up on their funds I'm saying these because I was a victim too After loosing 745,000 USD I lose my mind until I read about COIN HACK RECOVERY I decided to contact the company on: [email protected] and I'm glad I made the decision not to give up. they helped me to recover all my lost funds within two days.
08:04 Alena76

Most people have been scammed severally and they give up on their funds I'm saying these because I was a victim too After loosing 745,000 USD I lose my mind until I read about COIN HACK RECOVERY I decided to contact the company on: [email protected] and I'm glad I made the decision not to give up. they helped me to recover all my lost funds within two days.
08:15 Alena76

Most people have been scammed severally and they give up on their funds I'm saying these because I was a victim too After loosing 745,000 USD I lose my mind until I read about COIN HACK RECOVERY I decided to contact the company on: [email protected] and I'm glad I made the decision not to give up. they helped me to recover all my lost funds within two days.