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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

  2. Объяснение многослойного персептрона (MLP)

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

  4. Объяснение архитектуры KAN: Kolmogorov-Arnold Networks

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 = 1и p = 2 эта метрика соответствует манхэттенскому и евклидову расстояниям. Но нас интересует именно1 \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 множествеS:

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

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

В контекстеL^p- пространств, если ограничить меру стандартной мерой Лебега (например, длина, площадь, объём) на\mathbb{R}^n,то функцияf(x)принадлежитL^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 числоM, такое что расстояние между любой парой точек этого множества не превышаетM.

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

Например, множество (0, 1)^nограничено, но не замкнуто, так как не включает границы, а значит не компактно. А вот множество [0, 1]^nуже 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 функциюfиз 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 множествеK.То есть функцииf \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 пространстваL^pдля 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.

000944bd69b83f63e838fcd30489ae78.png

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 активацииf_{\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) = xЧто \ 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:

62efd833dde40d83ad4b122285667fea.gif

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}}965d045806a15d71f3a93ad3c130b13d.gif

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.

1f267ddc1ad946bb4421938631b750c4.gif

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, преобразуетN- 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 преобразованиеN-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 видf(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 преобразованиеF_{\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 симметричнымиK(\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 Kявляется линейным. По сути, данная модель производит линейное и последующее аффинное преобразование и является полностью линейной.

Рассмотрим набор из n 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:

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

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

Гдеn– 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 ядромf_{\text{activation}}(x) = xна квадратичнуюf_{\text{activation}}(x) = x ^2на 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 произвольномN- мерном пространстве.

И вот тут первая схема нам и пригодится:

\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}

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 степениn \in \mathbb{N}_0(неотрицательные целые числа) с параметром сдвигаr = 0определяется 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 размерностьMпревышает размерностьdдляd \geq 2иn \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 чиселx_1, x_2, \dots, x_dи натурального числаn:

(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}

гдеk_1, k_2, \dots, k_d \in \mathbb{N}_0,и суммаk_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^Mи 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 сMпри различныхdиn.

  • Еслиd \geq 2иn \geq 2:

    M = \binom{n + d - 1}{d - 1} = \frac{(n + d - 1)!}{n! \, (d - 1)!} >d

    При фиксированной размерностиdи увеличении степениn, Mрастёт полиномиально относительноn.

    При фиксированной степениnи увеличении размерностиd, Mрастёт экспоненциально относительноd.

  • Дляn = 1:

    M = d

    Полиномиальное ядро первой степени эквивалентно линейному ядру K_{\text{Linear}}и функция\varphi_{\text{poly }n}преобразует данные в исходное пространство признаков.

  • Дляd = 1:

    M=1

    Функция\varphi_{\text{poly }n}отображает входные данные в одномерное пространство признаков. Приn = 1эквивалент\varphi_{\text{linear }}.Приn\geq2производит 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 размерN \times M,гдеN– количество тренировочных векторов, аM– 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 будетN \times 3,для \varphi_{\text{poly }3}(x)– N \times 7для\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 естьf(\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!}

Подставляяz = \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 чиселx_1, x_2, \dots, x_dи натурального числаn:

(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}

гдеk_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 дляK_{\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^Mи 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:

\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}

Функция \varphi_\text{gaussian}(x) при\lambda = 2определяется 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}_0и 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}иM = 2.

j

\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 размеромN \times M, где N – 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 N.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:

\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}

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 ядроK_\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 пространствахL^{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}

Гдеc_{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 а и b 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 j 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| < 1и 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 j 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:

|z| < 1\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, доL-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 l-м слое.

  • \lambda^{(l)}_i– ширина ядра на l-м слое.

  • M_l– количество нейронов на l-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: ПлотностьS_Kв 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 семействоS_K,​состоящее из конечных линейных комбинаций сдвигов функцииK,

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 вL^p(\mathbb{R}^r)для любогоp \in [1, \infty).

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

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

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

Тогда семействоS_K,​состоящее из функций вида:

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

гдеM \in \mathbb{N},w_i \in \mathbb{R},иc_i \in \mathbb{R}^r,плотно вC(\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

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

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

иKне 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 2и 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 слоевL.

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 k слоями.

Теперь докажем, что утверждение верно дляL = 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 A,а функцию h: \mathbb{R} \rightarrow \mathbb{R} можно аппроксимировать с произвольной точностью функциями из семейства B,то композицияh \circ fможет быть аппроксимирована композициями функций из B и функций изA.

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 h равномерно непрерывна на значениях функции f,существуетL_h > 0такое, 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 f может быть аппроксимирована с точностью \delta сетью сk 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 изS_K​), то сеть с k + 1 слоями аппроксимирует f с точностью (L_h + 1) \delta.

Доказательство:

Пустьf_k– аппроксимация функции f сетью сkслоями, так что| f - f_k |_\infty < \delta.

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

Тогда h' \circ f_k аппроксимируетfс 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 f с \| f - f_i \|_\infty < \epsilon_iдляi = 1, \dots, N.

Рассмотрим линейную комбинацию:

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

Выбирая коэффициентыc_iсоответствующим 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 K 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 f 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 + 1слоями может аппроксимировать f 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 m являются универсальными аппроксиматорами для векторнозначных функций f: \mathbb{R}^r \rightarrow \mathbb{R}^m.

Так как каждая компонента f_j векторнозначной функции f может аппроксимироваться независимо многослойной RBF сетью, как показано в теореме 3. Объединяя аппроксимации всех компонент, получаем аппроксимацию функцииf 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_Lозначает, 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_b​​f_\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 подпространствоN- 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 d-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 еслиfявляется функцией нескольких перменных, тоf 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 G,который представляет шаг сетки аппроксимации. Для любых m \in [0, k],ошибка аппроксимацииf(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:

  • G – шаг сетки (чем меньше, тем точнее будет аппроксимация),

  • C – константа, зависящая от функции и её преобразований,

  • k – степень гладкости исходных функций \Phi,

  • m – 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 порядкаk+1на компактном множествеK– то есть вC^{k+1}(K).При этом множествоC^{k+1}(K)является подмножеством пространства непрерывных функцийC(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_b​f_\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 смещенияf_\text{bias}(x) = b,мы используем B-сплайныf_\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_2всё 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)

  1. 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]}

  2. 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 l = 0

    \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 l ={L-1}

    \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 l={L-1}

    \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 ограниченаM =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 вродеx_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 функцииf,количество 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 функциюfможно разложить на блоки\Phi,обладающие непрерывными производными порядкаk+1,и при этом константаCв 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.

5233de4fe464a5ad7cde36132fbe8bb0.png

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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  • 17.08.25 17:49 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

  • 18.08.25 21:10 Connorhelms

    I can honestly say losing my bitcoin savings was one of the darkest moments of my life. I had put everything I could into that investment, believing it would secure my future, only to wake up one day and realize the platform had vanished with all my funds. The feeling was indescribable, fear, shame, and helplessness all at once. I thought I’d never see that money again. Then I came across Alpha Spy Nest. At first, I was hesitant, I had already been deceived once, and the thought of trusting anyone else terrified me. But something about their professionalism and the way they patiently explained things gave me a little hope. For the first time in weeks, I felt like I wasn’t completely alone in the fight. They listened to my story without judgment, broke down the technical side in a way I could understand, and reassured me that there was still a chance. Watching them work was incredible. They traced every transaction, uncovered where my bitcoin had been moved, and kept me updated through every stage. It felt like they cared as much about my recovery as I did. The day they told me they had successfully recovered my lost savings, I actually cried. Seeing those funds back in my wallet felt unreal, it was like a huge weight had been lifted off my shoulders. Beyond just recovering my bitcoin, Alpha Spy Nest gave me back my peace of mind and restored a trust I thought I had lost forever. I’m forever grateful to them. If I hadn’t found Alpha Spy Nest, I’d still be stuck in regret and despair. Instead, I now share my story as proof that recovery is possible. You can also reach out to them via: Email: [email protected], WhatsApp: +15132924878‬, Telegram: https://t.me/Alphaspynest, Website: www.alphaspynest.org

  • 18.08.25 22:17 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

  • 18.08.25 22:17 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

  • 19.08.25 02:26 Jacksonbash

    You are not alone if you have had your bitcoins taken from your wallet or if you invested in an initial coin offering (ICO) that was a scam. I experienced the same thing. I first lost $173,000 to a fraudulent investment scam company in less than five months. I was able to get all of my money in five days thanks to a hacker my friend recommended. In order to raise awareness about these cryptocurrency criminals and do my part to minimize the number of victims, I'm speaking out. To report a victim, simply send an email to [email protected].

  • 19.08.25 18:15 keith

    RECOVER FROM CRYPTO SCAMS WITH THE HELP OF BONES RECOVERY BITCOIN EXPERT If someone needs to recover cryptocurrency that has been lost to wallet hackers, internet scammers, or BTC transferred to the wrong address, I suggest BONES RECOVERY BITCOIN, a very trustworthy recovery organization . Simply presenting my case to the expert and supplying the necessary data allowed BONES RECOVERY BITCOIN to assist me in recovering my Bitcoin. It's a relief that I recovered this much after losing considerably more to the fraudulent broker I initially put my trust in. Because we will inevitably make mistakes, we can never be diligent enough. If you have lost money to cryptocurrency scams, I strongly advise getting in touch with BONES RECOVERY BITCOIN. Contact address WhatsApp....‪+1-60-621-024-64‬ Email [email protected]

  • 19.08.25 18:54 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

  • 19.08.25 18:54 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

  • 19.08.25 18:54 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

  • 20.08.25 12:01 Official

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  • 20.08.25 17:00 lisared

    Few months ago I experienced one of the most devastating moments of my life €300k was stolen by hackers . When my money was stolen I felt hopeless, until I was introduced to Agent Jasmine Lopez her expertise and dedication gave me peace of mind. She skillfully traced complex transactions and tracked down the scammers, which would have been impossible for me to do on my own. Thanks to her tireless efforts, I was able to recover a significant portion of my lost funds. If you've fallen victim of hackers, I highly recommend reaching out to her. You can contact her via email at Recoveryfundprovider@gmail. com WhatsApp at +{44 736-644-5035}. Her attention to detail and commitment to her work are truly impressive. Don't hesitate to reach out her expertise could be the key to recovering your lost funds.

  • 20.08.25 20:42 vallatjosette

    My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com     OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number  +18737715701 Thanks.

  • 21.08.25 01:54 michaeldavenport218

    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

  • 21.08.25 01:54 michaeldavenport218

    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

  • 21.08.25 01:54 michaeldavenport218

    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

  • 21.08.25 14:56 elainemurray

    E.M.A.I.L. [email protected] W.h.a.t.s.A.p.p. +1(561)(726)(3697) It all began with an Instagram advertisement. A man presented himself as a cryptocurrency investment coach, showcasing a luxurious lifestyle and screenshots of substantial profits. He claimed his trading algorithm could double any investment in a week. Curious, I decided to reach out to him. He said the minimum investment was $50,000 and sent me a link to a platform that looked legitimate. At first, everything seemed promising. My balance increased steadily, charts showed growth, and I felt excited. When I tried to withdraw the money, the system blocked the transaction and demanded a release fee of $15,000. I paid it, but my account was soon marked under review, and the coach stopped responding. I realised I had been scammed. Desperate, I turned to Techy Force Cyber Retrieval. From the first consultation, they carefully reviewed my case and assured me the funds could be traced. They explained how they would track the wallet addresses and follow the path of the stolen money. Their clear guidance made me feel confident. Over the next few weeks, Techy Force Cyber Retrieval kept me updated on every step. They traced the transactions. Their persistence and expertise were obvious, and I could see real progress. Even when the process felt slow, Techy Force Cyber Retrieval maintained clear communication. They explained each action and why it mattered. Their careful attention made the ordeal manageable and reassuring. A month later, thanks to Techy Force Cyber Retrieval, I was able to recover 80% of my money and have it safely returned to my bank account. Relief and gratitude overwhelmed me. What started as a tempting opportunity had turned into a stressful ordeal, but Techy Force Cyber Retrieval transformed it into a successful resolution.

  • 21.08.25 21:17 monte5757

    I went through one of the most painful experiences of my life I lost over $850,000 to scammers. They had no mercy,things got so bad that I even sold some of my properties just to keep up with their demands. At one point, I felt completely hopeless and trapped. But everything changed when I came across a legit recovery agent who actually helped me recover my funds and return them safely to my wallet. It was like getting my life back. If you’ve ever been scammed, please don’t give up. Reach out to Recoveryfundprovider [at] gmail [dot] com WhatsApp at +{44 736-644-5035}.she’s professional, reliable, and truly knows how to get results. I’m genuinely grateful I found her, and I believe you’ll thank me later too.

  • 22.08.25 08:00 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

  • 22.08.25 08:00 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

  • 22.08.25 08:00 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

  • 23.08.25 05:11 ramiroalcazar

    A hacker skillfully impersonated one of our clients’ trusted suppliers and manipulated their finance department into transferring $180,000 worth of Bitcoin. The request appeared authentic, reinforced by falsified documents and meticulously crafted communication that mirrored the supplier’s established style. By the time the company uncovered the deception, the funds had already been funneled through numerous digital wallets. When they appealed to their bank for assistance, they were told recovery was impossible since cryptocurrency transactions are irreversible. The company suddenly faced the prospect of financial ruin. Confronted with this devastating loss, the client sought a specialized solution and turned to Techy Force Cyber Retrieval. From the initial consultation, our team demonstrated that hope remained. Unlike the bank, which dismissed their plea, Techy Force Cyber Retrieval meticulously analyzed every aspect of the fraud and assured the client that our blockchain specialists had the tools to trace and potentially reclaim the stolen assets. Our investigation began with a comprehensive audit of the fraudulent transactions. Using advanced blockchain forensics, Techy Force Cyber Retrieval’s analysts traced the stolen Bitcoin across a labyrinth of wallets and exchanges. Every development was documented with precision, and regular updates were delivered to the client, ensuring transparency and trust. This combination of technical mastery and clear communication provided reassurance during an otherwise overwhelming crisis. After persistent tracking and strategic intervention, Techy Force Cyber Retrieval achieved a remarkable outcome. From the original $180,000 that had been siphoned away, our team successfully recovered the equivalent of $160,000 in USDT. This restitution enabled the company to meet payroll obligations, stabilize its operations, and avert what had seemed like an inevitable collapse. What distinguished Techy Force Cyber Retrieval in the eyes of the client was not only our technical expertise but also our dedication and professionalism. We approached the case with urgency, maintained open communication at every stage, and treated the recovery effort with the utmost seriousness. At a moment when the client felt powerless, Techy Force Cyber Retrieval restored both their assets and their confidence. Now, victims of cryptocurrency fraud do not have to accept total loss. With the right expertise, recovery is achievable. Techy Force Cyber Retrieval stands as proof that even in situations deemed hopeless, we can deliver tangible results and provide businesses and individuals with a genuine second chance. CONSULT TECHY FORCE CYBER RETRIEVAL FOR A GREAT SOLUTION RECOVERY......... WHATSAPP.......... +15617263697 WEBSITE......... https://techyforcecyberretrieval.com

  • 24.08.25 01:05 samshaffer

    Have been on the quest for a way to recover my $195,000 from a forex broker. I was fortunate to meet Darek at ([email protected]) through a friend who was recommended as the best crypto recovery specialist he was the one who assisted me and restore my money back to my atomic wallet. I did reach him at [email protected] and it was an amazing experience. Never knew there was an expert who could trace back crypto until I met this great team. Don’t think twice, get connected with the right personnel now to avoid been fooled again..

  • 24.08.25 06:32 amy

    I have lost all hope to recover my investment in pragmatic trade which I lost $125,000 in the process of investment. I'm here to say thank you to anthony davies for being a man of your word got my funds recovered back plus a bonus within a week of working with [email protected] I hope people make use of this opportunity as well and don't give up on getting your funds back. is also available on Telegram: @anthonydavies01

  • 24.08.25 17:13 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

  • 24.08.25 17:13 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

  • 24.08.25 17:13 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

  • 24.08.25 18:39 nelly young

    Getting your USDC stolen feels like the ground has been pulled out from under you. It’s not just about the money you’ve been violated, and your security is at risk. And since crypto lives on the blockchain, there’s no bank to call and no quick refund. That’s why you need someone who knows exactly what to do. Agent Jasmine Lopez has helped countless victims trace stolen USDC and lock down their accounts before more damage happens. Time matters reach her today at [email protected] WhatsApp at +44 736-644-5035 and take the first step toward getting your funds and peace of mind back.

  • 24.08.25 23:56 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: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.08.25 23:56 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: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.08.25 23:56 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: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 25.08.25 00:55 vallatjosette

    My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com     OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number  +18737715701 Thanks.

  • 25.08.25 05:14 samshaffer

    Have been on the quest for a way to recover my $195,000 from a forex broker. I was fortunate to meet Darek at ([email protected]) through a friend who was recommended as the best crypto recovery specialist he was the one who assisted me and restore my money back to my atomic wallet. I did reach him at [email protected] and it was an amazing experience. Never knew there was an expert who could trace back crypto until I met this great team. Don’t think twice, get connected with the right personnel now to avoid been fooled again..

  • 26.08.25 10:14 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

  • 26.08.25 10:14 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

  • 26.08.25 10:14 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

  • 26.08.25 23:59 floridaberna

    It began as an entirely ordinary afternoon when an unexpected message appeared in my Facebook inbox. The sender was a woman I had never encountered. She introduced herself with warmth and courtesy, explaining that her late husband had been deeply invested in Bitcoin before his sudden passing. According to her narrative, he had left behind a vast fortune in cryptocurrency. However, due to intricate international banking regulations and legal entanglements, she was unable to access the funds without external assistance. Her words carried an air of urgency yet also sincerity. She claimed she had come across my profile in a cryptocurrency discussion forum and believed I might be the right person to help. The obstacle she faced was a set of legal fees amounting to $25,000. Once those costs were paid, she assured me the Bitcoin would be unlocked and a generous portion would be mine in gratitude. Against my better instincts, I allowed myself to be persuaded. She provided what appeared to be legitimate documents and even screenshots of supposed legal correspondence. The level of detail made her story feel authentic. After several exchanges, I transferred the $25,000.Almost immediately, her demeanor shifted. Responses became sporadic and vague. Within days, communication ceased entirely. My optimism dissolved into dread as I recognized the truth. I had been deceived. Refusing to surrender, I searched for solutions and discovered TECHY FORCE CYBER RETRIEVAL, a firm specializing in tracing and reclaiming stolen cryptocurrency. Their team outlined a meticulous plan to track the blockchain trail, identify the sequence of wallets involved, and coordinate with local authorities to freeze the assets before they disappeared into the labyrinth of the crypto underworld. Time was critical. In the volatile realm of digital currency, stolen funds can be laundered through countless transactions within hours. Yet TECHY FORCE CYBER RETRIEVAL pursued each lead with relentless precision. They provided regular progress updates and explained how every crypto transaction leaves a digital footprint, no matter how deeply it is buried. Weeks later, the breakthrough came. The fraudulent account was located, the illicit funds were flagged, and to my immense relief, I recovered 100% of my stolen money. This left me with an invaluable lesson. The internet is full of opportunists who prey on trust and desperation. But should you ever fall victim, know that there are experts like TECHY FORCE CYBER RETRIEVAL who can help you reclaim what is rightfully yours.  MAil     support (@) techyforcecyberretrieval (.) com        WhatsApp   +1 561 726 369 7         Website    ht tp s : / / techy force cyber retrieval . com

  • 27.08.25 17:24 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

  • 27.08.25 17:24 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

  • 27.08.25 17:24 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

  • 27.08.25 20:27 maximeden68

    Top Crypto Bitcoin Wallet Recovery Services Contact iFORCE HACKER RECOVERY I wish to express my sincere gratitude to iForce Hacker Recovery for their exceptional service in recovering my lost bitcoin. Their expertise and professionalism were evident throughout the process, providing me with reassurance during a challenging time. The team's dedication to client satisfaction and their effective strategies were instrumental in resolving my issue. I highly recommend their services to anyone facing similar challenges. Thank you, iForce Hacker Recovery. Contact iForce Hacker Recovery: Email: iforcehk @ consultant. com WhatsApp: +1 (240) 803-3706 Website: iforcehackers. com/

  • 27.08.25 21:21 dbeverly

    One morning I received an email that looked like it came from my cryptocurrency exchange, OKX. The subject warned: Suspicious activity detected and urged me to log in via a link. The message seemed authentic — perfect logos, formatting, even the sender’s address. My pulse spiked and I clicked. The site that opened was indistinguishable from OKX’s homepage. Convinced it was real, I entered my username and password. Moments later, dread set in: my account was emptied. $12,500 in Bitcoin was gone. I called OKX, hoping for a reversal, but the representative explained transfers were irreversible. Desperate, I turned to GRAYWARE TECH SERVICE. From the first call, their professionalism was clear. They carefully examined my case, tracing the stolen Bitcoin across the blockchain. They identified the destination wallets and mapped every move the hackers made. Their regular updates turned fear into cautious optimism. Acting swiftly, GRAYWARE coordinated with exchanges along the path of the stolen funds, freezing assets before they could vanish. Their vigilance and expertise kept me informed and reassured. After two tense weeks, the breakthrough came: my full $12,500 was recovered and returned to my OKX account. What began as a nightmare became a success thanks to GRAYWARE TECH SERVICE’s relentless determination. They didn’t just recover my money — they restored my confidence in digital security. Contact: WhatsApp: +1 858 275 9508 Website: https://graywaretechservice.com Email: [email protected]

  • 28.08.25 10:08 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

  • 28.08.25 10:08 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

  • 28.08.25 10:08 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

  • 28.08.25 13:50 Maureensels

    STEPS TO RECOVERING MONEY FROM A CRYPTO INVESTMENT SCAM CONTACT WIZARD LARRY RECOVERY EXPERTS Are you trying to find a means to get your other cryptocurrency, including Bitcoin, back from scammers? Stop searching! Call or text +1 (616) - (292) - (4789) or send a mail to (WIZARDLARRY(@)MAIL. C OM) is the greatest Scammed Crypto Recovery Agency I've seen on Google Earth. I'm delighted I made the brave decision to get in touch with them. In a matter of hours, all of the cryptocurrency that I had stolen was returned to my Coin Base wallet. For assistance with their strategies, I will advise you all to get in touch with their customer service. You will thank me later. Website... Wizardlarryrecovery. com Email [email protected] o m

  • 28.08.25 13:50 Maureensels

    STEPS TO RECOVERING MONEY FROM A CRYPTO INVESTMENT SCAM CONTACT WIZARD LARRY RECOVERY EXPERTS Are you trying to find a means to get your other cryptocurrency, including Bitcoin, back from scammers? Stop searching! Call or text +1 (616) - (292) - (4789) or send a mail to (WIZARDLARRY(@)MAIL. C OM) is the greatest Scammed Crypto Recovery Agency I've seen on Google Earth. I'm delighted I made the brave decision to get in touch with them. In a matter of hours, all of the cryptocurrency that I had stolen was returned to my Coin Base wallet. For assistance with their strategies, I will advise you all to get in touch with their customer service. You will thank me later. Website... Wizardlarryrecovery. com Email [email protected] o m

  • 28.08.25 16:31 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

  • 28.08.25 16:31 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

  • 28.08.25 19:17 leticiazavala

    Earlier this year, I found what looked like an unbeatable offer on first-class airline tickets through an online travel agency. They advertised heavily discounted fares for international travel for only 0.11 BTC, which was worth around 7300 USD at the time. The only condition was that payment had to be made in Bitcoin. The website looked professional, had convincing reviews, and even offered live chat support. After some hesitation, I decided to go ahead with the booking. Two days later, I received an email from the airline saying my reservation had been canceled. I went back to the agency’s website only to find it had disappeared. Their customer support was gone, and their social media accounts had vanished. That was when I realized I had been scammed. I thought my Bitcoin was lost for good. Unlike a credit card transaction, there was no way to reverse it. A friend recommended a company called Techy Force Cyber Retrieval that specializes in tracking and recovering stolen crypto. I reached out not expecting much. To my surprise, their team responded quickly. I sent them the transaction details, and they immediately began tracing the movement of the 0.11 BTC. They followed the funds through multiple wallets and discovered the Bitcoin had ended up at a large exchange known for cooperating in fraud investigations. Techy Force Cyber Retrieval worked fast. They submitted the evidence to the exchange and requested that the wallet be frozen. Over the next few weeks, they handled all communication and submitted the necessary documentation. Just under three weeks later, the exchange approved the release of my funds. I was stunned when I saw the 0.11 BTC back in my wallet. I had lost hope, but Techy Force Cyber Retrieval delivered. They were professional, honest, and transparent throughout the entire process. If you have been scammed in a crypto deal, do not assume the money is gone forever. With the right help, it is possible to trace and recover stolen assets. Techy Force Cyber Retrieval gave me a second chance, and I would not hesitate to recommend them to anyone who finds themselves in a similar situation. FOR SERVICE HELP Email: [email protected] [email protected] CALL +15617263697

  • 29.08.25 09:20 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

  • 29.08.25 09:20 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

  • 29.08.25 09:20 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

  • 30.08.25 01:40 Moises Velez

    In a single transaction, my Coinbase wallet was destroyed. I was at a loss as to what to do and was unable to contact Coinbase support Teams. This transaction resulted in the complete depletion of my Ethereum and Bitcoin. I was not the individual in question, and I required assistance after losing approximately $821,000. I visited the police station to submit a report, but they were also unable to provide any assistance. A gentleman approached me at the stiction and presented me with a note that included an email address (recoveryhacker101[at]gmail[dot]com). He informed me that this recovery company could be of assistance. Upon contacting them and submitting all necessary documentation within 72 hours, I received a letter stating that she had successfully recovered all of my funds and identified the individuals responsible for the criminal activity. I would like to inform everyone that it is feasible to retrieve their funds provided that they correspond with the appropriate individual and adhere to the prescribed procedures.

  • 31.08.25 14:19 jack118

    In 2018, Marie assisted in the recovery of stolen bitcoin. The service was excellent. The coins were stolen by hackers, creating a nightmare. Marie was suggested by a friend. Contact her at [email protected] and @MARIE_CONSULTANCY on Telegram or WhatsApp at +1 7127594675. She was responsible for recovering Bitcoin. The trading goes on.

  • 31.08.25 23:14 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

  • 31.08.25 23:14 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

  • 31.08.25 23:14 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

  • 01.09.25 04:55 Maurice Cherry

    Through Hackrontech recovery, Bitcoin scam victims can retrieve their money. I recommed hackrontech to anyone who has fallen victim to a scam and has been looking for methods and techniques to recover their lost cryptocurrency or wallets. Hackrontech Recovery is a reliable cryptocurrency recovery firm that assists victims in recovering their stolen cryptocurrency and offers secure solutions to protect your wallets from online scammers. I must admit that I was deeply melancholy and had given up on life until these experts could restore my 4.1 BTC to my wallet. If you’ve lost your cryptocurrency and you are helpless about it, contact hackrontech Recovery to get your money back. Email : hackrontech @gmail.com One key aspect that makes hackrontech Recovery stand out is its focus on providing secure solutions to protect wallets from online scammers. It’s not just about recovering lost funds; it’s also about preventing future incidents and ensuring that clients’ digital assets are safeguarded against potential threats. This proactive approach demonstrates their commitment to the long-term financial security of their clients. Furthermore, for individuals who have lost their cryptocurrency and are feeling helpless, reaching out to hackrontech Recovery could be a turning point in their situation. The reassurance that they are legitimate for seeking help and recovering lost funds can provide much-needed relief and a sense of empowerment. hackrontech Recovery as a reliable cryptocurrency recovery firm is certainly well-founded. Their ability to assist scam victims in recovering stolen cryptocurrency, their focus on providing secure solutions, and their commitment to supporting clients through challenging situations make them a valuable resource for individuals navigating the complex world of digital currencies. If you or someone you know has fallen victim to a cryptocurrency scam, contacting hackrontech Recovery could be the first step towards reclaiming lost funds and regaining peace of mind. Contact info; email: hackrontech @gmail.com

  • 01.09.25 11:06 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

  • 01.09.25 11:07 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

  • 01.09.25 16:41 Anitastar

    Wizard Hilton Cyber Tech's strategic approach to Bitcoin recovery represents a mutually beneficial collaboration that offers valuable advantages for all involved. At the core of this innovative partnership is a shared commitment to leveraging cutting-edge technology and expertise to tackle the increasingly complex challenges of cryptocurrency theft and loss. By combining Wizard Hilton's world-class cybersecurity capabilities with the deep industry insights and recovery methodologies of Cyber Tech, this alliance is poised to revolutionize the way individuals and organizations safeguard their digital assets. Through a meticulous, multi-layered process, the team meticulously analyzes each case, employing advanced forensic techniques to trace the flow of stolen funds and identify potential recovery avenues. This rigorous approach not only maximizes the chances of successful Bitcoin retrieval, but also provides invaluable intelligence to enhance future security measures and prevention strategies. Importantly, the collaboration is built on a foundation of trust, transparency, and a genuine concern for the well-being of clients, ensuring that the recovery process is handled with the utmost care and discretion. As the cryptocurrency landscape continues to evolve, this strategic alliance between Wizard Hilton Cyber Tech stands as a shining example of how industry leaders can come together to safeguard digital assets, protect victims of cybercrime, and pave the way for a more secure and resilient cryptocurrency ecosystem. Email : wizardhiltoncybertech ( @ ) gmail (. ) com     OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number  +18737715701 Thanks.

  • 01.09.25 17:03 WILLER GOODY

    Don’t be misled by the countless stories you come across online many of them are unreliable and can easily steer you in the wrong direction. I personally tried several so-called recovery services, only to end up frustrated and disappointed every single time. But everything changed when I finally connected with a true tech professional who genuinely knows what they’re doing. Instead of wasting time and energy with unskilled hackers who make empty promises, it’s far better to work with a proven expert who can actually recover your stolen or lost cryptocurrency, including Bitcoins. [email protected] is, without a doubt, one of the most trustworthy and skilled blockchain specialists I’ve ever come across. If you’ve lost money to scammers, don’t give up. Reach out to their email today and take the first step toward getting your coins back. You can also contact them at +44 736-644-5035.

  • 02.09.25 23:12 [email protected]

    Sylvester Bryant has recovered over £151,000 in lost cryptocurrency. You can reach him for assistance via email at Yt7cracker@gmail. com. For direct communication, Sylvester Bryant is available on WhatsApp at +1 512 577 7957. This service proved invaluable when other recovery methods had failed. The money was recovered quickly and effectively. My financial situation was dire prior to this. This recovery has restored my financial stability. This reliable support was crucial during a difficult time. Mr. Bryant's skill in retrieving lost funds is remarkable.

  • 03.09.25 07:17 Fabian3

    Scammers have ruined the forex trading market, they have deceived many people with their fake promises on high returns. I learnt my lesson the hard way. I only pity people who still consider investing with them. Please try and make your researches, you will definitely come across better and reliable forex trading agencies that would help you yield profit and not ripping off your money. Also, if your money has been ripped-off by these scammers, you can report to a regulated crypto investigative unit who make use of software to get money back. They helped me recover my scammed funds back to me If you encounter with some issue make sure you contact ([email protected]) they're recovery expert and a very effective and reliable .

  • 04.09.25 02:11 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

  • 04.09.25 02:11 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

  • 04.09.25 02:45 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

  • 04.09.25 16:24 Melbourne

    Sylvester Bryant has recovered over £151,000 in lost cryptocurrency. You can reach him for assistance via email at Yt7cracker@gmail. com. For direct communication, Sylvester Bryant is available on WhatsApp at +1 512 577 7957. This service proved invaluable when other recovery methods had failed. The money was recovered quickly and effectively. My financial situation was dire prior to this. This recovery has restored my financial stability. This reliable support was crucial during a difficult time. Mr. Bryant's skill in retrieving lost funds is remarkable.

  • 04.09.25 18:11 WILLER GOODY

    Don't fall for fake recovery scams online. Many people promise to help you get back lost crypto. They often just make excuses or don't deliver. This leads to more disappointment and lost hope. After many bad experiences, I finally found a real expert. They were professional and knew what they were doing. This person actually recovered my stolen coins. They gave me peace of mind back. If you lost crypto, contact them at [email protected] WhatsApp at +44 736-644-5035. Don't waste your time with others. They can help you get your money back.

  • 05.09.25 08:14 Amostampari

    How can I Recover My Lost Btc, Usdt - Wizard Larry Recovery Experts. Get Your Stolen Crypto Coins Back By Hiring A Hacker || I Can't Get Into My USDT? Account, It Seems Like I Was Hacked || Do You Need A Bitcoin Recovery Expert? Getting Lost, Stolen Crypto, or Hacked? How can I retrieve cryptocurrency from a con artist? Do I need a hacker to get my money back? Getting in dealing with a reputable business like Wizard Larry Recovery Experts could help you get your money back if you can't access your USDT account or have lost Bitcoin or NFT as a result of hacking or scams. For additional support and direction on your particular circumstance, it is advised that you get in touch with them at the address below ADDRESS Website... https://larrywizard43.wixsite.com/wizardlarry OR www.wizardlarryrecovery. com Email... wizardlarry@mail. com WhtsApp or call,.. +447 (311) 146 749

  • 05.09.25 15:39 micheal1219

    Losing USDT can be upsetting. This stablecoin usually stays close to the dollar's value. However, mistakes happen. Bugs in code, scams, or sending funds to the wrong place can cause losses. It might feel like the money is gone forever. But often, recovery is possible. Contact Marie at [email protected]. You can also reach her on Telegram at @marie_consultancy or WhatsApp at +17127594675.

  • 06.09.25 09:21 Young Felix

    [email protected] will Help you Recovery your Scammed Funds.

  • 06.09.25 09:22 Young Felix

    [email protected] will Help you Recovery your Scammed Funds. I just had to share my experience with Intel Fox Recovery Services. Initially, I was unsure if it would be possible to recover my BTC, 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. Here is another caution; not all recovery services are legit. I personally lost 3 BTC from my Binance account due to a deceptive platform. If you happen to have suffered a similar loss, consider INTEL FOX RECOVERY SERVICES. Their services not only showcase a highly mannerism of professionalism but also effectiveness of how they handle and assist their clients. Therefore, I highly recommend their services to anyone seeking assistance in recovering their stolen BTC.

  • 06.09.25 18:52 michaeldavenport218

    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

  • 06.09.25 18:52 michaeldavenport218

    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

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