Этот сайт использует файлы cookies. Продолжая просмотр страниц сайта, вы соглашаетесь с использованием файлов cookies. Если вам нужна дополнительная информация, пожалуйста, посетите страницу Политика файлов Cookie
Subscribe
Прямой эфир
Русский Русский  English English
Войти / Регистрация
Cryptocurrencies: 9056 / Markets: 115900
Market Cap: $ 3 125 122 318 352 / 24h Vol: $ 125 397 857 176 / BTC Dominance: 58.753780948153%

Н Новости

  1. Главная
  2. Новости

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

Источник

Теги

Категория

Новости

Дата

  • 11.11.25 19:28 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 11.11.25 19:29 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 12.11.25 04:09 [email protected]

    A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.

  • 12.11.25 04:10 [email protected]

    A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.

  • 12.11.25 05:07 peggy09

    Anthony Davies stands out in the world of crypto recovery. His skills come from years of hands-on work with blockchain tech. He helps people get back what thieves took, often against tough odds. i once lost $200000 to a fake broker site, anthony stood out as the only person who was able to get back my funds. you can contact him by sending an email to; anthonydaviestech @ gma1l com

  • 12.11.25 09:37 patricialovick86

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

  • 12.11.25 09:37 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

  • 13.11.25 19:01 peggy09

    i Lost $200,000 to a phishing scam in 2022. Funds went to a mixer service. Davies traced 70% through Ethereum layers. He teamed with an exchange to freeze the rest. i got $140,000 back in six months. Hiring Davies means clear steps. You share details. He checks facts. Then, the hunt begins. Expect ups and downs, but his plan keeps it steady. you can reach out to him by sending an email to anthonydaviestech {@} gmail com

  • 13.11.25 20:05 [email protected]

    A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.

  • 13.11.25 20:05 [email protected]

    A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.

  • 13.11.25 22:51 ashley11

    Recover All Lost Cryptocurrency From Scammers

  • 13.11.25 22:52 ashley11

    Recover All Lost Cryptocurrency From Scammers TREQORA INTEL has exhibited unparalleled strength in the realm of recovery. They stand out as the premier team to collaborate with if you encounter withdrawal difficulties from the platform where you’ve invested. Recently, I engaged with them to recover over a million dollars trapped in an investment platform I’d been involved with for months. I furnished their team with every detail of the investment via Email (SUPPORT @ TREQORA . C O M”), including accounts, names, and wallet addresses to which I sent the funds. This decision proved to be the best I’ve made, especially after realizing I had been sc**med by the company. Initially, I harbored doubts about their services, but I was proven wrong. TREQORA INTEL ensures exemplary service delivery and ensures the perpetrators face justice. They employ advanced techniques to ensure you regain access to your funds. Understandably, many individuals who have fallen victim to investment scams may still harbor trepidation about engaging in online services again due to the trauma of being sc**med. However, I implore you to take action. Seek assistance from TREQORA INTEL today and witness their remarkable capabilities firsthand. Among the myriad of hackers available, TREQORA INTEL stands head and shoulders above the rest. While I may not have sampled all of them, the few I attempted to work with previously were unhelpful and solely focused on depleting the little funds I had left. I am grateful that I resisted their enticements, and despite the time it took me to discover TREQORA INTEL, they ultimately fulfilled my primary objective. I am confident they will execute the task proficiently. Without their intervention, I would have remained despondent and perplexed indefinitely. Don’t make the error of entrusting sc**mers to rectify a sc*m; the consequences are evident. Email:support@treqora. com,WhatsApp: ‪‪‪‪‪‪+1 (7 7 3) 9 7 7 - 7 8 7 7‬‬‬‬ ‬‬,Website: Treqora. com. How Can I Recover My Lost Bitcoin From A Romance Scammer-HIRE TREQORA INTEL

  • 13.11.25 23:36 daisy

    i Lost $200,000 to a phishing scam in 2022. Funds went to a mixer service. Davies traced 70% through Ethereum layers. He teamed with an exchange to freeze the rest. i got $140,000 back in six months. Hiring Davies means clear steps. You share details. He checks facts. Then, the hunt begins. Expect ups and downs, but his plan keeps it steady. you can reach out to him by sending an email to anthonydaviestech {@} gmail com

  • 14.11.25 03:42 harristhomas7376

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

  • 14.11.25 03:42 harristhomas7376

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

  • 14.11.25 08:38 [email protected]

    A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.

  • 14.11.25 10:39 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $184,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez a crypto recovery agent. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 14.11.25 12:12 daisy

    i Lost $200,000 to a phishing scam in 2022. Funds went to a mixer service. Davies traced 70% through Ethereum layers. He teamed with an exchange to freeze the rest. i got $140,000 back in six months. Hiring Davies means clear steps. You share details. He checks facts. Then, the hunt begins. Expect ups and downs, but his plan keeps it steady. you can reach out to him by sending an email to anthonydaviestech {@} gmail com

  • 14.11.25 15:07 caridad

    Perth Family Saved After $400K Crypto Scam Our family in Perth, WA invested through what we thought was a trusted platform but ended up being a fraudulent investment scheme. We lost nearly AUD 420,000 worth of BTC and USDT. Luckily, a friend recommended Bitreclaim.com. Their 24/7 customer support assigned us a smart contract audit specialist who asked for wallet addresses and transaction hashes. With their forensic blockchain trace, they recovered over 5.1 BTC directly into our hardware wallet. For Perth investors: don’t give up hope. Submit a case at Bitreclaim.com immediately. Their professionalism and success rate in Australia is unmatched.

  • 14.11.25 18:33 justinekelly45

    FAST & RELIABLE CRYPTO RECOVERY SERVICES Hire iFORCE HACKER RECOVERY I was one of many victims deceived by fake cryptocurrency investment offers on Telegram. Hoping to build a retirement fund, I invested heavily and ended up losing about $470,000, including borrowed money. Just when I thought recovery was impossible, I found iForce Hacker Recovery. Their team of crypto recovery experts worked tirelessly and helped me recover my assets within 72 hours, even tracing the scammers involved. I’m deeply thankful for their professionalism and highly recommend their services to anyone facing a similar situation.  Website: ht tps://iforcehackers. co m WhatsApp: +1 240-803-3706   Email: iforcehk @ consultant. c om

  • 14.11.25 20:56 juliamarvin

    Firstly, the importance of verifying the authenticity of online communications, especially those about financial matters. Secondly, the potential for recovery exists even in cases where it seems hopeless, thanks to innovative services like TechY Force Cyber Retrieval. Lastly, the cryptocurrency community needs to be more aware of these risks and the available solutions to combat them. My experience serves as a warning to others to be cautious of online impersonators and never to underestimate the potential for recovery in the face of theft. It also highlights the critical role that professional retrieval services can play in securing your digital assets. In conclusion, while the cryptocurrency space offers unparalleled opportunities, it also presents unique challenges, and being informed and vigilant is key to navigating this landscape safely. W.h.a.t.s.A.p.p.. +.15.6.1.7.2.6.3.6.9.7. M.a.i.l T.e.c.h.y.f.o.r.c.e.c.y.b.e.r.r.e.t.r.i.e.v.a.l.@.c.o.n.s.u.l.t.a.n.t.c.o.m. T.e.l.e.g.r.a.m +.15.6.1.7.2.6.3.6.9.7

  • 15.11.25 12:47 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $184,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez a crypto recovery agent. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 15.11.25 14:39 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

  • 15.11.25 14:39 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

  • 15.11.25 15:31 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $184,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez a crypto recovery agent. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 15.11.25 15:52 [email protected]

    A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.

  • 16.11.25 14:43 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

  • 16.11.25 14:44 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

  • 16.11.25 20:38 [email protected]

    A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.

  • 17.11.25 03:24 johnny231

    INFO@THEBARRYCYBERINVESTIGATIONSDOTCOM is one of the best cyber hackers that i have actually met and had an encounter with, i was suspecting my partner was cheating on me for some time now but i was not sure of my assumptions so i had to contact BARRY CYBER INVESTIGATIONS to help me out with my suspicion. During the cause of their investigation they intercepted his text messages, social media(facebook, twittwer, snapchat whatsapp, instagram),also call logs as well as pictures and videos(deleted files also) they found out my spouse was cheating on me for over 3 years and was already even sending nudes out as well as money to anonymous wallets,so i deciced to file for a divorce and then when i did that i came to the understanding that most of the cryptocurrency we had invested in forex by him was already gone. BARRY CYBER INVESTIGATIONS helped me out through out the cause of my divorce with my spouse they also helped me in retrieving some of the cryptocurrency back, as if that was not enough i decided to introduce them to another of my friend who had lost her most of her savings on a bad crytpo investment and as a result of that it affected her credit score, BARRY CYBER INVESTIGATIONS helped her recover some of the funds back and helped her build her credit score, i have never seen anything like this in my life and to top it off they are very professional and they have intergrity to it you can contact them also on their whatsapp +1814-488-3301. for any hacking or pi jobs you can contact them and i assure you nothing but the best out of the job

  • 17.11.25 11:26 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

  • 17.11.25 11:27 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

  • 19.11.25 01:56 VERONICAFREDDIE809

    Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I got connected to the best female expert AGENT Jasmine Lopez,,( [email protected] ) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.

  • 19.11.25 08:11 JuneWatkins

    I’m June Watkins from California. I never thought I’d lose my life savings in Bitcoin. One wrong click, a fake wallet update, and $187,000 vanished in seconds. I cried for days, felt stupid, ashamed, and completely hopeless. But God wouldn’t let me stay silent or defeated. A friend sent me a simple message: “Contact Mbcoin Recovery Group, they specialize in this.” I was skeptical (there are so many scammers), but something in my spirit said “try.” I reached out to Mbcoin Recovery Group through their official site and within minutes their team responded with kindness and clarity. They walked with me step by step, and stayed in constant contact. Three days later, I watched in tears as every single Bitcoin returned to my wallet, 100% recovered. God turned my mess into a message and my shame into a testimony! If you’ve lost crypto and feel it’s gone forever, don’t give up. I’m living proof that recovery is possible. Thank you, Mbcoin Recovery Group, and thank You, Jesus, for never leaving me stranded. contact: (https://mbcoinrecoverygrou.wixsite.com/mb-coin-recovery) (Email: [email protected]) (Call Number: +1 346 954-1564)

  • 19.11.25 08:26 elizabethmadison

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

  • 19.11.25 08:27 elizabethmadison

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

  • 19.11.25 16:30 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

  • 19.11.25 16:30 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

  • 20.11.25 15:55 mariotuttle94

    HIRE THE BEST HACKER ONLINE FOR CRYPTO BITCOIN SCAM RECOVERY / iFORCE HACKER RECOVERY After a security breach, my husband lost $133,000 in Bitcoin. We sought help from a professional cybersecurity team iForce Hacker Recovery they guided us through each step of the recovery process. Their expertise allowed them to trace the compromised funds and help us understand how the breach occurred. The experience brought us clarity, restored a sense of stability, and reminded us of the importance of strong digital asset and security practices.  Website: ht tps:/ /iforcehackers. c om WhatsApp: +1 240-803-3706 Email: iforcehk @ consultant. c om

  • 21.11.25 10:56 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: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 21.11.25 10:56 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: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 22.11.25 04:41 VERONICAFREDDIE809

    Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I got connected to the best female expert AGENT Jasmine Lopez,,( [email protected] ) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.

  • 22.11.25 22:04 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

  • 22.11.25 22:04 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

  • 22.11.25 22:05 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

  • 23.11.25 03:34 Matt Kegan

    SolidBlock Forensics are absolutely the best Crypto forensics team, they're swift to action and accurate

  • 23.11.25 09:54 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 23.11.25 18:01 mosbygerry

    I recently had the opportunity to work with a skilled programmer who specialized in recovering crypto assets, and the results were nothing short of impressive. The experience not only helped me regain control of my investments but also provided valuable insight into the intricacies of cryptocurrency technology and cybersecurity. The journey began when I attempted to withdraw $183,000 from an investment firm, only to encounter a series of challenges that made it impossible for me to access my funds. Despite seeking assistance from individuals claiming to be Bitcoin miners, I was unable to recover my investments. The situation was further complicated by the fact that all my deposits were made using various cryptocurrencies that are difficult to trace. However, I persisted in my pursuit of recovery, driven by the determination to reclaim my losses. It was during this time that I discovered TechY Force Cyber Retrieval, a team of experts with a proven track record of successfully recovering crypto assets. With their assistance, I was finally able to recover my investments, and in doing so, gained a deeper understanding of the complex mechanisms that underpin cryptocurrency transactions. The experience taught me that with the right expertise and guidance, even the most seemingly insurmountable challenges can be overcome. I feel a sense of obligation to share my positive experience with others who may have fallen victim to cryptocurrency scams or are struggling to recover their investments. If you find yourself in a similar situation, I highly recommend seeking the assistance of a trustworthy and skilled programmer, such as those at TechY Force Cyber Retrieval. WhatsApp (+1561726 3697) or (+1561726 3697). Their expertise and dedication to helping individuals recover their crypto assets are truly commendable, and I have no hesitation in endorsing their services to anyone in need. By sharing my story, I hope to provide a beacon of hope for those who may have lost faith in their ability to recover their investments and to emphasize the importance of seeking professional help when navigating the complex world of cryptocurrency.

  • 24.11.25 11:43 michaeldavenport238

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

  • 24.11.25 11:43 michaeldavenport238

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

  • 24.11.25 16:34 Mundo

    I wired 120k in crypto to the wrong wallet. One dumb slip-up, and poof gone. That hit me hard. Lost everything I had built up. Crypto moves on the blockchain. It's like a public record book. Once you send, that's it. No take-backs. Banks can fix wire mistakes. Not here. Transfers stick forever. a buddy tipped me off right away. Meet Sylvester Bryant. Guy's a pro at pulling back lost crypto. Handles cases others can't touch, he spots scammer moves cold. Follows money down secret paths. Mixers. Fake trades. Hidden swaps. You name it, he tracks it. this happens to tons of folks. Fat-finger a key. Miss one digit in the address. Boom. Billions vanish like that each year. I panicked. Figured my stash was toast for good. Bryant flipped the script. He jumps on hard jobs quick. Digs deep. Cracks the trail. Got my funds back safe. You're in the same boat? Don't sit there. Hit him up today. Email [email protected]. WhatsApp +1 512 577 7957. Or +44 7428 662701. Time's your enemy here. Scammers spend fast. Chains churn non-stop. Move now. Grab your cash back home.

  • 25.11.25 05:15 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

  • 25.11.25 13:31 mickaelroques52

    CRYPTO TRACING AND INVESTIGATION EXPERT: HOW TO RECOVER STOLEN CRYPTO_HIRE RAPID DIGITAL RECOVERY

  • 25.11.25 13:31 mickaelroques52

    I’ve always considered myself a careful person when it comes to money, but even the most cautious people can be fooled. A few months ago, I invested some of my Bitcoin into what I believed was a legitimate platform. Everything seemed right, professional website, live chat support and even convincing testimonials. I thought I had done my homework. But when I tried to withdraw my funds, everything fell apart. My account was blocked, the so-called support team disappeared and I realized I had been scammed. The shock was overwhelming. I couldn’t believe I had fallen for it. That Bitcoin represented years of savings and sacrifices and it felt like everything had been stolen from me in seconds. I didn’t sleep for days and I was angry at myself for trusting the wrong people. In my desperation, I started searching for solutions and came across Rapid Digital Recovery. At first, I thought it was just another promise that would lead nowhere. But after speaking with them, I realized this was different. They were professional, clear and understanding. They explained exactly how they track stolen funds through blockchain forensics and what steps would be taken in my case. I gave them all the transaction details and they immediately got to work. What impressed me most was their transparency, they gave me updates regularly and kept me involved in the process. After weeks of investigation, they achieved what I thought was impossible: they recovered my stolen Bitcoin and safely returned it to my wallet. The relief I felt that day is indescribable. I went from feeling hopeless and broken to feeling like I had been given a second chance. I am forever grateful to Rapid Digital Recovery. They didn’t just recover my money, they restored my peace of mind. If you’re reading this because you’ve been scammed, please know you’re not alone and that recovery is possible. I’m living proof that with the right help, you can get your funds back... Contact Info Below WhatSapp:  + 1 414 807 1485 Email:  rapiddigitalrecovery (@) execs. com Telegram:  + 1 680 5881 631

  • 26.11.25 18:18 harristhomas7376

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

  • 26.11.25 18:20 harristhomas7376

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

  • 26.11.25 19:13 James Robert

    I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have  access to  my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site

  • 26.11.25 19:13 James Robert

    I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have  access to  my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site

  • 26.11.25 19:13 James Robert

    I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have  access to  my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site

  • 27.11.25 10:56 harristhomas7376

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

  • 27.11.25 10:56 harristhomas7376

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

  • 27.11.25 20:04 deborah113

    Scammed Crypto Asset Recovery Solution Hire iFORCE HACKER RECOVERY When I traded online, I lost both my investment money and the anticipated gains.  Before permitting any withdrawals, the site kept requesting more money, and soon I recognized I had been duped.  It was really hard to deal with the loss after their customer service ceased responding.  I saw a Facebook testimonial about how iForce Hacker Recovery assisted a victim of fraud in getting back the bitcoin she had transferred to con artists.  I contacted iForce Hacker Recovery, submitted all relevant case paperwork, and meticulously followed the guidelines.  I'm relieved that I was eventually able to get my money back, including the gains that were initially displayed on my account. I'm sharing my story to let others who have been conned know that you can recover your money. WhatsApp: +1 240-803-3706 Email: iforcehk @ consultant. c om Website: ht tps:/ /iforcehackers. c om

  • 27.11.25 23:48 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 27.11.25 23:48 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 28.11.25 00:08 VERONICAFREDDIE809

    Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected])WhatsApp at +44 736-644-5035. ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it.

  • 28.11.25 11:15 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

  • 28.11.25 11:15 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

  • 28.11.25 11:43 elizabethmadison

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

  • 28.11.25 11:43 elizabethmadison

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

  • 29.11.25 12:35 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 29.11.25 12:35 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 29.11.25 12:35 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 30.11.25 20:37 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

  • 01.12.25 12:27 Thomas Muller

    YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with,  website https://greatwhiprecoveryc.wixsite.com/greatwhip-site  text +1(406)2729101 email [email protected]

  • 01.12.25 12:27 Thomas Muller

    YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with,  website https://greatwhiprecoveryc.wixsite.com/greatwhip-site  text +1(406)2729101 email [email protected]

  • 01.12.25 23:45 michaeldavenport238

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

  • 01.12.25 23:45 michaeldavenport238

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

  • 02.12.25 02:21 donald121

    In 2025 alone, hackers stole over $1.5 billion in digital assets from users worldwide. That's a wake-up call for anyone holding crypto. Theft hits hard because once funds move, they're tough to get back. Common ways it happens include phishing emails that trick you into giving up keys, big exchange breaches, or malware sneaking into your wallet. Marie guide walks you through steps to recover stolen cryptocurrency. You'll learn quick actions to stop more loss, how to trace funds, and ways to fight back legally. Plus, tips to avoid this mess next time. reach her (infocyberrecoveryinc@gmail com and whatsapp:+1 7127594675)

  • 02.12.25 15:05 Matt Kegan

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

  • 03.12.25 09:22 tyrelldavis1

    I still recall the day I fell victim to an online scam, losing a substantial amount of money to a cunning fraudster. The feeling of helplessness and despair that followed was overwhelming, and I thought I had lost all hope of ever recovering my stolen funds. However, after months of searching for a solution, I stumbled upon a beacon of hope - GRAYWARE TECH SERVICE, a highly reputable and exceptionally skilled investigative and recovery firm. Their team of expert cybersecurity professionals specializes in tracking and retrieving money lost to internet fraud, and I was impressed by their unwavering dedication to helping victims like me. With their extensive knowledge and cutting-edge technology, they were able to navigate the complex world of online finance and identify the culprits behind my loss. What struck me most about GRAYWARE TECH SERVICE was their unparalleled expertise and exceptional customer service. They took the time to understand my situation, provided me with regular updates, and kept me informed throughout the entire recovery process. Their transparency and professionalism were truly reassuring, and I felt confident that I had finally found a reliable partner to help me recover my stolen money. Thanks to GRAYWARE TECH SERVICE, I was able to recover a significant portion of my lost funds, and I am forever grateful for their assistance. Their success in retrieving my money not only restored my financial stability but also restored my faith in the ability of authorities to combat online fraud. If you have fallen victim to internet scams, I highly recommend reaching out to GRAYWARE TECH SERVICE - their expertise and dedication to recovering stolen funds are unparalleled, and they may be your only hope for retrieving what is rightfully yours. You can reach them on whatsapp+18582759508 web at ( https://graywaretechservice.com/ )    also on Mail: ([email protected]

  • 03.12.25 21:01 VERONICAFREDDIE809

    Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected]) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.

  • 03.12.25 22:17 Tonerdomark

    I lost $300,000 in USDC to a phishing scam. Scammers tricked me with a fake wallet link. They drained my account fast. I felt hopeless. No way to get it back. Then Sylvester stepped in. His skills traced the funds. He recovered every bit. USDC is a stablecoin tied to the dollar. Phishing scams hit hard in crypto. They fool you with urgent emails or sites. Billions vanish each year this way. Sylvester knows blockchain tracks. He used tools to follow the trail. I got my money back in weeks. Skills like his turn loss to win. Don't wait if scammed. Contact Mr. Sylvester now. Email: yt7cracker@gmail. com. WhatsApp only: + 1 512 577 7957 or + 44 7428 662701. He helped me. He can help you.

  • 04.12.25 01:37 michaeldavenport238

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

  • 04.12.25 01:37 michaeldavenport238

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

  • 04.12.25 04:35 Tonerdomark

    I lost $300,000 in USDC to a phishing scam. Scammers tricked me with a fake wallet link. They drained my account fast. I felt hopeless. No way to get it back. Then Sylvester stepped in. His skills traced the funds. He recovered every bit. USDC is a stablecoin tied to the dollar. Phishing scams hit hard in crypto. They fool you with urgent emails or sites. Billions vanish each year this way. Sylvester knows blockchain tracks. He used tools to follow the trail. I got my money back in weeks. Skills like his turn loss to win. Don't wait if scammed. Contact Mr. Sylvester now. Email: [email protected]. WhatsApp only: + 1 512 577 7957 or + 44 7428 662701. He helped me. He can help you.

  • 04.12.25 10:32 Tonerdomark

    I lost $300,000 in USDC to a phishing scam. Scammers tricked me with a fake wallet link. They drained my account fast. I felt hopeless. No way to get it back. Then Sylvester stepped in. His skills traced the funds. He recovered every bit. USDC is a stablecoin tied to the dollar. Phishing scams hit hard in crypto. They fool you with urgent emails or sites. Billions vanish each year this way. Sylvester knows blockchain tracks. He used tools to follow the trail. I got my money back in weeks. Skills like his turn loss to win. Don't wait if scammed. Contact Mr. Sylvester now. Email: [email protected]. WhatsApp only: + 1 512 577 7957 or + 44 7428 662701. He helped me. He can help you.

  • 04.12.25 18:25 smithhazael

    Hire Proficient Expert Consultant For any form of lost crypto "A man in Indonesia tragically took his own life after losing his family's savings to a scam. The shame and blame were too much to bear. It's heartbreaking to think he might still be alive if he knew help existed. "PROFICIENT EXPERT CONSULTANTS, I worked alongside PROFICIENT EXPERT CONSULTANTS when I lost my funds to an investment platform on Telegram. PROFICIENT EXPERT CONSULTANTS did a praiseworthy job, tracked and successfully recovered all my lost funds a total of $770,000 within 48hours after contacting them, with their verse experience in recovery issues and top tier skills they were able to transfer back all my funds into my account, to top it up I had full access to my account and immediately converted it to cash, they handled my case with professionalism and empathy and successfully recovered all my lost funds, with so many good reviews about PROFICIENT EXPERT CONSULTANTS, I’m glad I followed my instincts after reading all the reviews and I was able to recovery everything I thought I had lost, don’t commit suicide if in any case you are caught in the same situation, contact: Proficientexpert@consultant. com Telegram: @ PROFICIENTEXPERT, the reliable experts in recovery.

  • 04.12.25 21:45 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 04.12.25 21:45 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 05.12.25 08:35 into11

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

  • 05.12.25 08:48 Tonerdomark

    SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]

  • 06.12.25 01:44 Tonerdomark

    SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]

  • 06.12.25 01:48 Tonerdomark

    SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]

  • 06.12.25 10:36 Thomas Muller

    YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with,  website https://greatwhiprecoveryc.wixsite.com/greatwhip-site  text +1(406)2729101 email [email protected]

  • 06.12.25 10:36 Thomas Muller

    YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with,  website https://greatwhiprecoveryc.wixsite.com/greatwhip-site  text +1(406)2729101 email [email protected]

  • 06.12.25 10:39 michaeldavenport238

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

  • 06.12.25 10:42 michaeldavenport238

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

  • 07.12.25 08:43 Tonerdomark

    SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]

  • 02:17 liam

    I recently fell a victim of cryptocurrency investment and mining scam, I lost almost all my life savings to BTC scammers. I almost gave up because the amount of crypto I lost was too much. So I spoke to a friend who told me about ANTHONYDAVIESTECH company. I Contacted them through their email and i provided them with the necessary information they requested from me and they told me to be patient and wait to see the outcome of their job. I was shocked after two days my Bitcoin was returned to my Wallet. All thanks to them for their genius work. I Contacted them via Email: anthonydaviestech @ gmail . com all thanks to my friend who saved my life

  • 09:07 Tonerdomark

    SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]

Для участия в Чате вам необходим бесплатный аккаунт pro-blockchain.com Войти Регистрация

Похожие новости

Есть вопросы?
С вами на связи 24/7
Поддержка 24/7 Help Icon
Менеджер онлайн Help Icon
Help Icon