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Н 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

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

1.1 Вкратце о некоторых дополнительных концепциях

Векторные пространства и подпространства

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

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

Аффинные пространства и подпространства

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

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

Лебеговы пространства

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

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

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

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

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

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

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

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

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

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

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

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

Кроме того, можно определить расстояние между функциями:

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

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

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

Компактные множества

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

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

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

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

2. Объяснение многослойного персептрона (MLP)

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

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

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

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

Теперь рассмотрим некоторые из них:

Теорема Цыбенко (1989)

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

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

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

Есть также пример ограниченной ширины и глубины:

Майоров и Пинкус (1999)

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

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

Теорема Хорника 1989 и Теорема Хорника 1991

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

Расширение Лешно и соавторов (1992):

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

2.1 Иллюстрация на примере

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

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

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

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

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

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

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

И также посмотрим на график:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Рассмотрим на примере который мы выше использовали.

Аффинное преобразование:

При аффинном преобразовании элементы результатирующего вектора будут иметь следующий вид :

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

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

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

Функция активации:

Преобразование с помощью функции активации будут иметь следующий вид:

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

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

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

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

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

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

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

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

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

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

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

3.1 SVM как вид нейронной сети прямого распространения

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

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

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

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

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

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

Часто используемыми подобными ядрами являются:

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

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

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

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

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

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

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

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

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

Рассмотрим набор из тренировочных векторов:

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

Также рассмотрим набор из тестовых векторов:

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

Формулу нашей модели можно тогда подать как:

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

Для наглядности можно записать матрицы в развернутом виде:

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

Можно также переписать и саму формулу модели:

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

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

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

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

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

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

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

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

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

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

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

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

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

3.1.1 Полиномиальное ядро

Возьмем вторую схему:

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

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

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

Где– показатель степени полиномиального ядра.

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

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

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

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

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

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

И вот тут первая схема нам и пригодится:

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

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

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

Вот доказательство:

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

И иллюстрация этого преобразования:

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

Для трёхмерных векторов с полиномиальным ядром второй степени:

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

Для двумерных векторов с полиномиальным ядром третьей степени:

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

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

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

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

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

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

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

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

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

Подставляя это разложение обратно в выражение для ядра, получаем:

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

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

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

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

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

Теперь проанализируем что происходит спри различныхи

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

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

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

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

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

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

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

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

3.1.2 Гауссово ядро

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

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

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

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

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

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

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

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

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

Ряд Тейлора для экспоненты выглядит следующим образом:

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

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

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

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

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

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

В нашем случае:

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

Применяя мультиномиальную теорему, получаем:

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

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

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

Упрощая, получаем:

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

Мы знаем, что:

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

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

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

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

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

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

Возьмем Гауссово ядро:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Теперь построим матрицу преобразования, используя тренировочные векторы:

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

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

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

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

Тогда формула SVM будет выглядеть следующим образом:

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

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

3.2 RBF нейронная сеть

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

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

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

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

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

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

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

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

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

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

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

И сейчас я приведу для каждого доказательство:

Inverse Quadratic RBF

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

где и равны:

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

Приведение к форме, удобной для разложения:

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

При условии, что:

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

Можем использовать разложение в геометрическую прогрессию:

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

Таким образом, получаем:

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

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

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

{}где:

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

Здесь:

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

Вычисление скалярного произведения:

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

Inverse Multiquadric RBF

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

Тогда ядро можно записать как:

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

Как и в доказательстве в предыдущем ядре:

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

Тогда ядро переписывается как:

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

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

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

Теперь наша цель – разложить выражение в степенной ряд:

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

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

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

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

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

где:

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

Здесь:

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

Скалярное произведение функций неявного преобразования будет:

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

Проверка сходимости разложения

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

Условие сходимости геометрической прогрессии для обоих ядер:

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

По неравенству Коши-Буняковского:

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

Наша задача — показать, что:

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

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

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

Заметим, что:

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

Так как неравенство является справедливым:

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

То выполняется и данное условие:

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

3.2.1 Многослойная RBF нейронная сеть

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

Для первого слоя:

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

Для второго слоя:

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

И так далее, до-го слоя:

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

Здесь:

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

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

Теорема универсальной аппроксимации многослойной RBF нейронной сети

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

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

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

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

Тогда семейство S_K, состоящее из конечных линейных комбинаций сдвигов функции

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

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

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

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

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

Тогда семейство S_K, состоящее из функций вида:

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

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

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

Теорема 3: Универсальная аппроксимация многослойными RBF сетями

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

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

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

Доказательство:

Докажем это методом математической индукции по числу слоев.

Базовый случай ( L = 2 ):

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

Индуктивный шаг:

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

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

Лемма 3.1: Композиция универсальных аппроксиматоров

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

Доказательство:

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

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

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

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

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

Лемма 3.2: Композиция в многослойных сетях

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

Доказательство:

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

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

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

Лемма 3.3: Линейная комбинация универсальных аппроксиматоров

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

Доказательство:

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

Рассмотрим линейную комбинацию:

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

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

Завершение индуктивного шага:

По индукционному предположению существует f_k такая, что

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

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

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

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

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

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

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

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

Следствие 3.1: Универсальная аппроксимация векторнозначных функций

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

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

Следствие 3.2: Универсальная аппроксимация в пространствах

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

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

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

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

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

Реализация многослойной RBF нейронной сети на PyTorch.

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

SkipConection MRBFN

import torch
import torch.nn as nn

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

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

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

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

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

    def reset_parameters(self):

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

    def forward(self, input):

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

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

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

        return self.basis_func(distances)

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

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

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

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

        else:
            self.skip_connection = nn.Identity()

    def forward(self, x):

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

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

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

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

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

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

    def forward(self, x):

        x1 = self.rbf_block1(x)

        x2 = self.rbf_block2(x1)

        x3 = self.rbf_block3(x2)

        return self.linear_final(x3)

Dense MRBFN

import torch
import torch.nn as nn

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

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

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

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

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

    def reset_parameters(self):

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

    def forward(self, input):

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

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

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

        return self.basis_func(distances)

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

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

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

    def forward(self, x, previous_outputs):

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

        rbf_densenet_output = self.rbf(combined_inputs)

        return rbf_densenet_output

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

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

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

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

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


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

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

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

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

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

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

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

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

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

        return self.linear_final(x3)

4. Объяснение архитектуры KAN: Kolmogorov-Arnold Networks

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

KAN – семейство нейронных сетей прямого распространения имеющих следующий вид:

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

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

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

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

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

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

Нейронная сеть будет иметь следующий вид:

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

Где каждый слой представляет собой преобразование подобного рода:

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

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

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

4.1 MLP как частный случай KAN

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

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

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

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

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

Это идентично KAN с двумя слоями:

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

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

Первый слой:

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

И второй выходной:

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

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

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

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

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

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

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

Первый слой:

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

Второй слой (выходной):

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

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

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

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

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

4.2 Теорема Колмогорова — Арнольда

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

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

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

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

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

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

4.3 B-spline KAT и B-spline KAN

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

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

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

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

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

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

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

где:

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

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

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

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

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

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

4.4 SVM и RBF нейронная сеть как частный случай KAN

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

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

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

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

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

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

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

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

Возьмем простой и привычный пример для двумерных входных данных:

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

Нам было бы очень интересно подать в виде:

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

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

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

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

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

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

Существует вариант разложить подобным образом:

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

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

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

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

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

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

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

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

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

4.4.1 Многослойная RBF нейронная сеть (MRBFN) vs FastKAN

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

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

Архитектура FastKAN выглядит следующим образом:

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

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

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

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

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

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

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

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

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

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

Начнем с:

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

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

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

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

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

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

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

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

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

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

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

4.5 Проклятье и благословение размерности

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4.6 Итог и обобщение информации с помощью множеств

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

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

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

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

Так что будет интересно!

Источники

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

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

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

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

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

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

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

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

KAN: Kolmogorov-Arnold Networks. Ziming Liu et al

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

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

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

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

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

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

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

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5 янв. 2025 г.

12.11.24 00:50 TERESA

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17.11.24 09:31 Vivianlocke223

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19.11.24 03:06 [email protected]

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19.11.24 03:07 [email protected]

My entire existence fell apart when a malevolent hacker recently gained access to my online accounts. I felt violated and extremely uneasy after discovering that the digital platforms I depended on for communication, employment, and finances had been compromised. Regaining control and restoring my digital security was an overwhelming task in the immediate aftermath. To help me navigate the difficult process of recovering my accounts and getting my peace of mind back, TRUST GEEKS HACK EXPERT came into my life as a ray of hope. They immediately put their highly skilled professionals to work, thoroughly examining the vulnerability and methodically preventing unwanted access. They guided me through each stage soothingly, explaining what was occurring and why, so I never felt lost or alone. They communicated with service providers to restore my legitimate access while skillfully navigating the complex labyrinth of account recovery procedures. My digital footprint was cleaned and strengthened against future attacks thanks to their equally amazing ability to remove any remaining evidence of the hacker's presence. However, TRUST GEEKS HACK EXPERT actual worth went beyond its technical aspects. They offered constant emotional support during the ordeal, understanding my fragility and sense of violation. My tense nerves were calmed by their comforting presence and kind comments, which served as a reminder that I wasn't alone in this struggle. With their help, I was able to reestablish my sense of security and control, which enabled me to return my attention to the significant areas of my life that had been upended. Ultimately, TRUST GEEKS HACK EXPERT all-encompassing strategy not only recovered my online accounts but also my general peace of mind, which is a priceless result for which I am incredibly appreciative of their knowledge and kindness. Make the approach and send a message to TRUST GEEKS HACK EXPERT Via Web site <> www://trustgeekshackexpert.com/-- E>mail: Trustgeekshackexpert(At)fastservice..com -- TeleGram,<> Trustgeekshackexpert
21.11.24 04:14 ronaldandre617

Being a parent is great until your toddler figures out how to use your devices. One afternoon, I left my phone unattended for just a few minutes rookie mistake of the century. I thought I’d take a quick break, but little did I know that my curious little genius was about to embark on a digital adventure. By the time I came back, I was greeted by two shocking revelations: my toddler had somehow managed to buy a $5 dinosaur toy online and, even more alarmingly, had locked me out of my cryptocurrency wallet holding a hefty $75,000. Yes, you heard that right a dinosaur toy was the least of my worries! At first, I laughed it off. I mean, what toddler doesn’t have a penchant for expensive toys? But then reality set in. I stared at my phone in disbelief, desperately trying to guess whatever random string of gibberish my toddler had typed as a new password. Was it “dinosaur”? Or perhaps “sippy cup”? I felt like I was in a bizarre game of Password Gone Wrong. Every attempt led to failure, and soon the laughter faded, replaced by sheer panic. I was in way over my head, and my heart raced as the countdown of time ticked away. That’s when I decided to take action and turned to Digital Tech Guard Recovery, hoping they could solve the mystery that was my toddler’s handiwork. I explained my predicament, half-expecting them to chuckle at my misfortune, but they were incredibly professional and empathetic. Their confidence put me at ease, and I knew I was in good hands. Contact With WhatsApp: +1 (443) 859 - 2886 Email digital tech guard . com Telegram: digital tech guard recovery . com website link :: https : // digital tech guard . com Their team took on the challenge like pros, employing their advanced techniques to unlock my wallet with a level of skill I can only describe as magical. As I paced around, anxiously waiting for updates, I imagined my toddler inadvertently locking away my life savings forever. But lo and behold, it didn’t take long for Digital Tech Guard Recovery to work their magic. Not only did they recover the $75,000, but they also gave me invaluable tips on securing my wallet better like not leaving it accessible to tiny fingers! Who knew parenting could lead to such dramatic situations? Crisis averted, and I learned my lesson: always keep my devices out of reach of little explorers. If you ever find yourself in a similar predicament whether it’s tech-savvy toddlers or other digital disasters don’t hesitate to reach out to Digital Tech Guard Recovery. They saved my funds and my sanity, proving that no challenge is too great, even when it involves a toddler’s mischievous fingers!
21.11.24 08:02 Emily Hunter

If I hadn't found a review online and filed a complaint via email to support@deftrecoup. com , the people behind this unregulated scheme would have gotten away with leaving me in financial ruins. It was truly the most difficult period of my life.
22.11.24 04:41 [email protected]

I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
22.11.24 15:26 cliftonhandyman

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22.11.24 23:43 teresaborja

all thanks to Tech Cyber Force Recovery expert assistance. As a novice in cryptocurrency, I had been carefully accumulating a modest amount of Bitcoin, meticulously safeguarding my digital wallet and private keys. However, as the adage goes, the best-laid plans can often go awry, and that's precisely what happened to me. Due to a series of technical mishaps and human errors, I found myself locked out of my Bitcoin wallet, unable to access the fruits of my digital labors. Panic set in as I frantically searched for a solution, scouring the internet for any glimmer of hope. That's when I stumbled upon the Tech Cyber Force Recovery team, a group of seasoned cryptocurrency specialists who had built a reputation for their ability to recover lost or inaccessible digital assets. Skeptical at first, I reached out, desperate for a miracle. To my utter amazement, the Tech Cyber Force Recovery experts quickly assessed my situation and devised a meticulous plan of attack. Through their deep technical knowledge, unwavering determination, and a keen eye for detail, they were able to navigate the complex labyrinth of blockchain technology, ultimately recovering my entire Bitcoin portfolio. What had once seemed like a hopeless endeavor was now a reality, and I found myself once again in possession of my digital wealth, all thanks to the incredible efforts of the Tech Cyber Force Recovery team. This experience has not only restored my faith in the cryptocurrency ecosystem. Still, it has also instilled in me a profound appreciation for the critical role that expert recovery services can play in safeguarding one's digital assets. ENAIL < Tech cybers force recovery @ cyber services. com > WEBSITE < ht tps : // tech cyber force recovery. info > TEXT < +1. 561. 726. 3697 >
24.11.24 02:21 [email protected]

I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
25.11.24 02:19 briankennedy

COMMENT ON I NEED A HACKER TO RECOVER MONEY FROM BINARY TRADING. HIRE FASTFUND RECOVERY
25.11.24 02:20 briankennedy

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26.11.24 21:59 [email protected]

In a world brimming with enticing investment opportunities, it is crucial to tread carefully. The rise of digital currencies has attracted many eager investors, but along with this excitement lurk deceitful characters ready to exploit the unsuspecting. I learned this lesson the hard way, and I want to share my story in the hopes that it can save someone from making the same mistakes I did. It all began innocently enough when I came across an engaging individual on Facebook. Lured in by promises of high returns in the cryptocurrency market, I felt the electric thrill of potential wealth coursing through me. Initial investments returned some profits, and that exhilarating taste of success fueled my ambition. Encouraged by a meager withdrawal, I decided to commit even more funds. This was the moment I let my guard down, blinded by greed. As time went on, the red flags started to multiply. The moment I tried to withdraw my earnings, a cascade of unreasonable fees appeared like a thick mist, obscuring the truth. “Just a little more,” they said, “Just until the next phase.” I watched my hard-earned money slip through my fingers as I scraped together every last cent to pay those relentless fees. My trust had become my downfall. In the end, I lost not just a significant amount of cash, but my peace of mind about $1.1 million vanished into the abyss of false promises and hollow guarantees. But despair birthed hope. After a cascade of letdowns, I enlisted the help of KAY-NINE CYBER SERVICES, a team that specializes in reclaiming lost funds from scams. Amazingly, they worked tirelessly to piece together what had been ripped away, providing me with honest guidance when I felt utterly defeated. Their expertise in navigating the treacherous waters of crypto recovery was a lifeline I desperately needed. To anyone reading this, please let my story serve as a warning. High returns often come wrapped in the guise of deception. Protect your investments, scrutinize every opportunity, and trust your instincts. Remember, the allure of quick riches can lead you straight to heartbreak, but with cautious determination and support, it is possible to begin healing from such devastating loss. Stay informed, stay vigilant, and may you choose your investment paths wisely. Email: kaynine @ cyberservices . com
26.11.24 23:12 rickrobinson8

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26.11.24 23:12 rickrobinson8

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27.11.24 00:39 [email protected]

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27.11.24 09:10 Michal Novotny

The biggest issue with cryptocurrency is that it is unregulated, wh ich is why different people can come up with different fake stories all the time, and it is unfortunate that platforms like Facebook and others only care about the money they make from them through ads. I saw an ad on Facebook for Cointiger and fell into the scam, losing over $30,000. I reported it to Facebook, but they did nothing until I discovered deftrecoup . c o m from a crypto community; they retrieved approximately 95% of the total amount I lost.
01.12.24 17:21 KollanderMurdasanu

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02.12.24 23:02 ytre89

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04.12.24 22:24 andreygagloev

When I first heard about Bitcoin back in 2018, I was skeptical. The idea of a decentralized, digital currency seemed too good to be true. But I was intrigued as I learned more about the technology behind it and its potential. I started small, investing just a few hundred dollars, dipping my toes into the cryptocurrency waters. At first, it was exhilarating to watch the value of my investment grow exponentially. I felt like I was part of the future, an early adopter of this revolutionary new asset. But that euphoria was short-lived. One day, I logged into my digital wallet only to find it empty - my Bitcoin had vanished without a trace. It turned out that the online exchange I had trusted had been hacked, and my funds were stolen. I was devastated, both financially and emotionally. All the potential I had seen in Bitcoin was tainted by the harsh reality that with decentralization came a lack of regulation and oversight. My hard-earned money was gone, lost to the ether of the digital world. This experience taught me a painful lesson about the price of trust in the uncharted territory of cryptocurrency. While the technology holds incredible promise, the risks can be catastrophic if you don't approach it with extreme caution. My Bitcoin investment gamble had failed, and I was left to pick up the pieces, wiser but poorer for having placed my faith in the wrong hands. My sincere appreciation goes to MUYERN TRUST HACKER. You are my hero in recovering my lost funds. Send a direct m a i l ( muyerntrusted ( @ ) mail-me ( . )c o m ) or message on whats app : + 1 ( 4-4-0 ) ( 3 -3 -5 ) ( 0-2-0-5 )
12.12.24 00:35 amandagregory

HOW TO HIRE A HACKER TO RECOVER STOLEN BITCOIN WITH FASTFUND RECOVERY
12.12.24 00:35 amandagregory

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27.12.24 00:21 swiftdream

I lost about $475,000.00 USD to a fake cryptocurrency trading platform a few weeks back after I got lured into the trading platform with the intent of earning a 15% profit daily trading on the platform. It was a hell of a time for me as I could hardly pay my bills and got me ruined financially. I had to confide in a close friend of mine who then introduced me to this crypto recovery team with the best recovery SWIFTDREAM i contacted them and they were able to completely recover my stolen digital assets with ease. Their service was superb, and my problems were solved in swift action, It only took them 48 hours to investigate and track down those scammers and my funds were returned to me. I strongly recommend this team to anyone going through a similar situation with their investment or fund theft to look up this team for the best appropriate solution to avoid losing huge funds to these scammers. Send complaint to Email: info [email protected]
31.12.24 04:53 Annette_Phillips

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04.01.25 04:56 florencestella

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04.01.25 04:57 florencestella

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06.01.25 19:09 michaeljordan15

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18.01.25 12:41 michaeldavenport218

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18.01.25 12:41 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] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected]
20.01.25 15:39 patricialovick86

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22.01.25 21:43 DoraJaimes23

Recovery expert. I lost my bitcoin to fake blockchain impostors on Facebook, they contacted me as blockchain official support and i fell stupidly for their mischievous act, this made them gain access into my blockchain wallet whereby 7.0938 btc was stolen from my wallet in total .I was almost in a comma and dumbfounded because this was all my savings i relied on . Then I made a research online and found a recovery expert , with the contact address- { RECOVERYHACKER101 (@) GMAIL . COM }... I wrote directly to the specialist explaining my loss. Hence, he helped me recover a significant part of my investment just after 2 days he helped me launch the recovery program , and the culprits were identified as well , all thanks to his expertise . I hope I have been able to help someone as well . Reach out to the recovery specialist to recover you lost funds from any form of online scam Thanks
23.01.25 02:36 [email protected]

After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
23.01.25 02:37 [email protected]

After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
23.01.25 14:20 nellymargaret

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26.01.25 03:54 [email protected]

Losing access to my crypto wallet account was one of the most stressful experiences ever. After spending countless hours building up my portfolio, I suddenly found myself locked out of my account without access. To make matters worse, the email address I had linked to my wallet was no longer active. When I tried reaching out, I received an error message stating that the domain was no longer in use, leaving me in complete confusion and panic. It was as though everything I had worked so hard for was gone, and I had no idea how to get it back. The hardest part wasn’t just the loss of access it was the feeling of helplessness. Crypto transactions are often irreversible, and since my wallet held significant investments, the thought that my hard-earned money could be lost forever was incredibly disheartening. I spent hours scouring forums and searching for ways to recover my funds, but most of the advice seemed either too vague or too complicated to be of any real help. With no support from the wallet provider and my email account out of reach, I was left feeling like I had no way to fix the situation.That’s when I found out about Trust Geeks Hack Expert . I was hesitant at first, but after reading about their expertise in recovering lost crypto wallets, I decided to give them a try. I reached out to their team, and from the very beginning, they were professional, understanding, and empathetic to my situation. They quickly assured me that there was a way to recover my wallet, and they got to work immediately.Thanks to Trust Geeks Hack Expert , my wallet and funds were recovered, and I couldn’t be more grateful. The process wasn’t easy, but their team guided me through each step with precision and care. The sense of relief I felt when I regained access to my crypto wallet and saw my funds safely back in place was indescribable. If you find yourself in a similar situation, I highly recommend reaching out to Trust Geeks Hack Expert. contact Them through EMAIL: [email protected] + WEBSITE. HTTPS://TRUSTGEEKSHACKEXPERT.COM + TELE GRAM: TRUSTGEEKSHACKEXPERT
28.01.25 21:48 [email protected]

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02.02.25 20:53 Michael9090

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05.02.25 00:04 Jannetjeersten

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06.02.25 19:42 Marta Golomb

My name is Marta, and I’m sharing my experience in the hope that it might help others avoid a similar scam. A few weeks ago, I received an email that appeared to be from the "Department of Health and Human Services (DHS)." It claimed I was eligible for a $72,000 grant debit card, which seemed like an incredible opportunity. At first, I was skeptical, but the email looked so professional and convincing that I thought it might be real. The email instructed me to click on a link to claim the grant, and unfortunately, I followed through. I filled out some personal details, and then, unexpectedly, I was told I needed to pay a "processing fee" to finalize the grant. I was hesitant, but the urgency of the message pushed me to make the payment, believing it was a necessary step to receive the funds. Once the payment was made, things quickly went downhill. The website became unreachable, and I couldn’t get in touch with anyone from the supposed DHS. It soon became clear that I had been scammed. The email, which seemed so legitimate, had been a clever trick to steal my money.Devastated and unsure of what to do, I began searching for ways to recover my lost funds. That’s when I found Tech Cyber Force Recovery, a team of experts who specialize in tracing stolen money and assisting victims of online fraud. They were incredibly reassuring and quickly got to work on my case. After several days of investigation, they managed to track down the scammers and recover my funds. I can’t express how grateful I am for their help. Without Tech Cyber Force Recovery, I don’t know what I would have done. This experience has taught me a valuable lesson: online scams are more common than I realized, and the scammers behind them are incredibly skilled. They prey on people’s trust, making it easy to fall for their tricks. HOW CAN I RECOVER MY LOST BTC,USDT =Telegram= +1 561-726-36-97 =WhatsApp= +1 561-726-36-97
08.02.25 05:45 [email protected]

I'm incredibly grateful that I did enough research to recover my stolen cryptocurrency. When I first fell victim to a scam, I felt hopeless and lost, unsure if I'd ever see my funds again. A few months ago, I was approached by someone on Telegram who claimed to have a lucrative investment opportunity in cryptocurrencies. They promised huge returns and played on my emotions, making it seem like a can't-miss chance. I was so eager to make my money grow that I didn't fully vet the situation, and unfortunately, I ended up falling for the scam. They guided me to invest a significant amount of money, and soon after, I realized I had been duped. The scammers blocked me, and my funds were gone. I felt devastated. All of my savings had been wiped out in what seemed like an instant, and the feeling of being taken advantage of was crushing. I spent days researching how to recover my stolen cryptocurrency but found the process to be overwhelming and complicated. I was starting to lose hope when I came across Trust Geeks Hack Expert. At first, I was skeptical about reaching out to a cryptocurrency recovery company, but after reading testimonials and researching their reputation, I decided to give them a try. I contacted Trust Geeks Hack Expert Website: www://trustgeekshackexpert.com/, and I was immediately reassured by their professionalism and expertise. They took the time to listen to my situation, and they were honest about what could and could not be done. What stood out to me was their deep understanding of cryptocurrency fraud and the recovery process. They were able to track down the scammers and initiate the recovery of my stolen funds, step by step. Thanks to Trust Geeks Hack Expert, I was able to get back a significant portion of the cryptocurrency I had lost. Their team was responsive, transparent, and diligent in their efforts. I was kept informed throughout the entire process, and they made sure I felt supported every step of the way. I truly can't thank them enough for their dedication and for restoring my faith in the possibility of recovery after such a devastating loss. I will definitely recommend Trust Geeks Hack Expert to anyone who has fallen victim to a cryptocurrency scam. TeleGram: Trustgeekshackexpert & what's A p p +1 7 1 9 4 9 2 2 6 9 3
08.02.25 05:46 [email protected]

I'm incredibly grateful that I did enough research to recover my stolen cryptocurrency. When I first fell victim to a scam, I felt hopeless and lost, unsure if I'd ever see my funds again. A few months ago, I was approached by someone on Telegram who claimed to have a lucrative investment opportunity in cryptocurrencies. They promised huge returns and played on my emotions, making it seem like a can't-miss chance. I was so eager to make my money grow that I didn't fully vet the situation, and unfortunately, I ended up falling for the scam. They guided me to invest a significant amount of money, and soon after, I realized I had been duped. The scammers blocked me, and my funds were gone. I felt devastated. All of my savings had been wiped out in what seemed like an instant, and the feeling of being taken advantage of was crushing. I spent days researching how to recover my stolen cryptocurrency but found the process to be overwhelming and complicated. I was starting to lose hope when I came across Trust Geeks Hack Expert. At first, I was skeptical about reaching out to a cryptocurrency recovery company, but after reading testimonials and researching their reputation, I decided to give them a try. I contacted Trust Geeks Hack Expert Website: www://trustgeekshackexpert.com/, and I was immediately reassured by their professionalism and expertise. They took the time to listen to my situation, and they were honest about what could and could not be done. What stood out to me was their deep understanding of cryptocurrency fraud and the recovery process. They were able to track down the scammers and initiate the recovery of my stolen funds, step by step. Thanks to Trust Geeks Hack Expert, I was able to get back a significant portion of the cryptocurrency I had lost. Their team was responsive, transparent, and diligent in their efforts. I was kept informed throughout the entire process, and they made sure I felt supported every step of the way. I truly can't thank them enough for their dedication and for restoring my faith in the possibility of recovery after such a devastating loss. I will definitely recommend Trust Geeks Hack Expert to anyone who has fallen victim to a cryptocurrency scam. TeleGram: Trustgeekshackexpert & what's A p p +1 7 1 9 4 9 2 2 6 9 3
10.02.25 21:22 sulabhakuchchal

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11.02.25 04:24 heyemiliohutchinson

I invested substantially in Bitcoin, believing it would secure my future. For a while, things seemed to be going well. The market fluctuated, but I was confident my investment would pay off. But catastrophe struck without warning. I lost access to my Bitcoin holdings as a result of several technical issues and inadequate security measures. Every coin in my wallet suddenly disappeared, leaving me with an overpowering sense of grief. The emotional impact of this loss was far greater than I had imagined. I spiraled into despair, feeling as though my dreams of financial independence were crushed. I was on the verge of giving up when I came across Assets_Recovery_Crusader. Being willing to give them a chance, I had nothing left to lose. They listened to my narrative and took the time to comprehend the particulars of my circumstance, rather than treating me like a case number. They worked diligently, using their advanced recovery techniques and deep understanding of blockchain technology to track down my lost Bitcoin. Assets_Recovery_Crusader rebuilt my trust in the bitcoin space. The financial impact had a significant emotional toll, but I was able to get past it thanks to Assets_Recovery_Crusader’s proficiency and persistence. For proper talks, reach out to them via TELEGRAM : Assets_Recovery_Crusader EMAIL: [email protected]
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13.02.25 14:45 aoifewalsh130

TELEGRAM: u/BestwebwizardRecovery EMAIL: [email protected] WEBSITE: https://bestwebwizrecovery.com/ The money I invested was meant for something incredibly important—my wedding. After months of saving, I had finally accumulated enough to make the day truly special. Wanting to grow this fund, I came across a crypto site called Abcfxb.pro, which promised daily returns through “AI crypto arbitrage trading.” They claimed they could deliver 1% returns on my investment every day, and I saw this as an opportunity to multiply my savings quickly. I thought it was the perfect way to ensure I’d have enough to cover all the wedding expenses. For the first few days, everything seemed perfect. I saw the promised returns and was able to withdraw money without any issues. It felt like a legitimate opportunity, and I was excited as my wedding fund grew. However, things took a turn when I tried to withdraw again. The site claimed that my account balance had fallen below their liquidity requirement and asked me to deposit more funds to proceed. Reluctantly, I deposited more money, believing it was just a minor issue. But the situation only worsened. I was then told that my withdrawal would take 50 days due to “blockchain congestion.” I wasn’t too concerned at first, thinking it was just a delay. But after 50 days, I still hadn’t received my funds, and they gave me the same excuse. Desperate, I contacted the site again, only to be informed that I would need to pay a 15% fee for “technical support” from the “Federal Reserve’s blockchain regulator” before I could withdraw my money. By now, I realized I had fallen victim to a scam. As I researched further, I found that others had been scammed in the same way, and the scammers had moved to another site with nearly the same layout. It was then that I came across a review from another victim, who explained how Best Web Wizard Recovery had helped him recover his lost funds. Desperate for a solution, I reached out to Best Web Wizard Recovery. To my relief, they responded quickly and professionally. Within six hours, they had successfully recovered my full investment. I was beyond grateful, especially since the money had been intended for my wedding. Thanks to their help, I was able to not only get my money back but also go ahead with my wedding as planned. It was a day I will always cherish, and I owe it to Best Web Wizard Recovery for helping me make it a reality. I highly recommend their services to anyone who has fallen victim to a crypto scam.
13.02.25 16:50 andytom798

I lost $210,000 worth of Bitcoin to a group of fake blockchain impostors on Red note, a Chinese app. They contacted me, pretending to be official blockchain support, and I was misled into believing they were legitimate. At the time, I had been saving up in Bitcoin, hoping to take advantage of the rising market. The scammers were convincing, and I made the mistake of trusting them with access to my blockchain wallet. To my shock and disbelief, they stole a total of $10,000 worth of Bitcoin from my wallet. It was devastating, as this amount represented all of my hard-earned savings. I was in utter disbelief, feeling foolish for falling for their deceptive tactics. I felt lost, as though everything I had worked towards was taken from me in an instant. Thankfully, my uncle suggested I reach out to an expert in cryptocurrency recovery. After doing some research online, I came across CYBERPOINT RECOVERY COMPANY. I was hesitant at first, but their positive reviews gave me some hope. I decided to contact them directly and explained my situation, including the amount I had lost and how the scammers had gained access to my account. To my relief, the team at CYBERPOINT RECOVERY responded quickly and assured me they could help. They launched a detailed recovery program, using advanced tools and techniques to trace the stolen Bitcoin. Within a matter of days, they successfully recovered my full $210,000 worth of Bitcoin, and they even identified the individuals behind the scam. Their expertise and professionalism made a huge difference, and I was incredibly grateful for their support. If you find yourself in a similar situation, I highly recommend reaching out to Cyber Constable Intelligence. They helped me recover my funds when I thought all hope was lost. Whether you’ve lost money to scammers or any other form of online fraud, they have the knowledge and resources to help you get your funds back. Don’t give up there are experts who can help you reclaim what you’ve lost. I’m sharing my story to hopefully guide others who are going through something similar. Here's Their Info Below ([email protected]) or W.H.A.T.S.A.P.P:+1.7.6.0.9.2.3.7.4.0.7
13.02.25 16:52 birenderkumar20101

I was able to reclaim my lost Bitcoin assets worth of $480,99 which i had lost to the scam company known as Capitalix fx a scam company pretending to be an investment platform which alot of people including myself have lost their funds to, sadly not all would be fortunate enough to retrieve back their funds like I did but if you’re reading this today then you’re already a step closer towards regaining your lost digital assets, CYBERPOINT RECOVERY COMPANY successfully retrieved back my funds in less than of 48hours after I sought for their help to get back my funds. This experience has taught me the importance of carrying out my due diligence before embracing any financial opportunity presented to me and while risk taking may be a part of the journey, some risks are not worth taking and never again will I involve myself with any online financial investment. It’s only right that we seek for external intervention and support of a higher knowledge system when it comes to digital assets recovery, Get in contact today with the team to get started on Email: ([email protected])
13.02.25 17:37 eunice49954

Agent Jasmine Lopez focuses on recovering stolen cryptocurrency, particularly USDT. She is well-known for helping victims of digital asset theft. Her reputation arises from successful recoveries that have allowed many to regain their lost funds. I witnessed this when $122,000 was taken from me. Thanks to Ms. Lopez's skills, I recovered the entire amount in just 24 hours. Her prompt response and effective methods relieved my financial burden. Ms. Lopez’s commitment to helping others is evident. She is always available to offer solutions to those facing similar problems. For assistance, she can be reached via email at recoveryfundprovider@gmail . com or contacted directly on WhatsApp and text at +44 - 7366 445035. Her Instagram handle is recoveryfundprovider.
14.02.25 02:50 Vladimir876

I was able to reclaim my lost Bitcoin assets worth of $480,99 which i had lost to the scam company known as Capitalix fx a scam company pretending to be an investment platform which alot of people including myself have lost their funds to, sadly not all would be fortunate enough to retrieve back their funds like I did but if you’re reading this today then you’re already a step closer towards regaining your lost digital assets, CYBERPOINT RECOVERY COMPANY successfully retrieved back my funds in less than of 48hours after I sought for their help to get back my funds. This experience has taught me the importance of carrying out my due diligence before embracing any financial opportunity presented to me and while risk taking may be a part of the journey, some risks are not worth taking and never again will I involve myself with any online financial investment. It’s only right that we seek for external intervention and support of a higher knowledge system when it comes to digital assets recovery, Get in contact today with the team to get started on Email: ([email protected]) or W.H.A.T.S.A.P.P:+1.7.6.0.9.2.3.7.4.0.7
14.02.25 02:56 christophadelbert3

Я был в полном смятении, когда потерял все свои сбережения, инвестируя в криптовалюту. Со мной связалась онлайн женщина по электронной почте, выдавая себя за менеджера по работе с клиентами банка, которая сказала мне, что я могу удвоить свои сбережения, инвестируя в криптовалюту. Я никогда не думал, что это будет мошенничество, и я потеряю все. Это продолжалось неделями, пока я не понял, что меня обманули. Вся надежда была потеряна, я был опустошен и разорен, к счастью для меня, я наткнулся на статью в моем местном бюллетене о CYBERPUNK RECOVERY Bitcoin Recovery. Я связался с ними и предоставил всю информацию по моему делу. Я был поражен тем, как быстро они вернули мои криптовалютные средства и смогли отследить этих мошенников. Я действительно благодарен за их услуги и рекомендую CYBERPUNK RECOVERY всем, кому нужно вернуть свои средства. Настоятельно рекомендую вам связаться с CYBERPUNK, если вы потеряли свои биткойны USDT или ETH из-за инвестиций в биткойны Электронная почта: ([email protected]) W.h.a.t.s.A.p.p (+.1.7.6.0.9.2.3.7.4.0.7)
14.02.25 02:56 christophadelbert3

I was in total dismay when I lost my entire savings investing in cryptocurrency, I was contacted online by a lady through email pretending to be an account manager of a bank, who told me I could make double my savings through cryptocurrency investment, I never imagined it would be a scam and I was going to lose everything. It went on for weeks until I realized that I have been scammed. All hope was lost, I was devastated and broke, fortunately for me, I came across an article on my local bulletin about CYBERPUNK RECOVERY Bitcoin Recovery, I contacted them and provided all the information regarding my case, I was amazed at how quickly they recovered my cryptocurrency funds and was able to trace down those scammers. I’m truly grateful for their service and I recommend CYBERPUNK RECOVERY to everyone who needs to recover their funds urge you to contact CYBERPUNK if you have lost your bitcoin USDT or ETH through bitcoin investment Email: ([email protected]) WhatsApp (+17609237407)
14.02.25 15:33 prelogmilivoj

I never imagined I would find myself in a situation where I was scammed out of such a significant amount of money, but it happened. I became a victim of a fake online donation project that cost me over $30,000. It all started innocently enough when I was searching for assistance after a devastating fire incident in California. While looking for support, I came across an advertisement that seemed to offer donations for fire victims. The ad appeared legitimate, and I reached out to the project manager to inquire about how to receive the donations. The manager was very convincing and insisted that in order to qualify for the donations, I needed to pay $30,000 upfront. In return, I was promised $1 million in donations. It sounded a bit too good to be true, but in my desperate situation, I made the mistake of believing it. The thought of receiving a substantial amount of help to rebuild after the fire clouded my judgment, and I went ahead and sent the money. However, after transferring the funds, the promised donations never arrived, and the manager disappeared. That’s when I realized I had been scammed. Feeling lost, helpless, and completely betrayed, I tried everything I could to contact the scammer, but all my efforts were in vain. Desperation led me to search for help online, hoping to find a way to recover my money and potentially track down the scammer. That’s when I stumbled upon several testimonies from others who had fallen victim to similar scams and had been helped by a company called Tech Cyber Force Recovery. I reached out to them immediately, providing all the details of the scam and the information I had gathered. To my immense relief, the experts at Tech Cyber Force Recovery acted swiftly. Within just 27 hours, they were able to locate the scammer and initiate the recovery process. Not only did they help me recover the $30,000 I had lost, but the most satisfying part was that the scammer was apprehended by local authorities in their region. Thanks to Tech Cyber Force Recovery, I was able to get my money back and hold the scammer accountable for their actions. I am incredibly grateful for their professionalism, expertise, and dedication to helping victims like me. If you have fallen victim to a scam or fraudulent activity, I highly recommend contacting Tech Cyber Force Recovery. They provide swift and efficient recovery assistance, and I can confidently say they made all the difference in my situation. ☎☎ 1️⃣5️⃣6️⃣1️⃣7️⃣2️⃣6️⃣3️⃣6️⃣9️⃣7️⃣ ☎☎ 📩 1️⃣5️⃣6️⃣1️⃣7️⃣2️⃣6️⃣3️⃣6️⃣9️⃣7️⃣ 📩
14.02.25 22:12 eunice49954

Agent Jasmine Lopez focuses on recovering stolen cryptocurrency, particularly USDT. She is well-known for helping victims of digital asset theft. Her reputation arises from successful recoveries that have allowed many to regain their lost funds. I witnessed this when $122,000 was taken from me. Thanks to Ms. Lopez's skills, I recovered the entire amount in just 24 hours. Her prompt response and effective methods relieved my financial burden. Ms. Lopez’s commitment to helping others is evident. She is always available to offer solutions to those facing similar problems. For assistance, she can be reached via email at recoveryfundprovider@gmail . com or contacted directly on WhatsApp and text at +44 - 7366 445035. Her Instagram handle is recoveryfundprovider.
15.02.25 02:51 Michelle Lynn

Living in Los Angeles, I never imagined I’d face such a difficult chapter in my life. At the time, my wife was pregnant, and we were both excited about starting a family. I fell victim to a series of scams, losing over $170,000 in total. Just when I thought things couldn’t get worse, I received a call from someone who promised to help me recover my losses. Desperate to fix the situation, I went along with it, hoping for a breakthrough. But it turned out to be another scam. However, most of the options I found either seemed dubious or offered no real guarantees. That’s when I came across Cyber Constable Intelligence. It was a company recommended in a Facebook community The team worked tirelessly on my case, and after some time, they successfully recovered 99% of my investment. Although I didn’t recover everything, the 99% recovery was a huge relief They also educated me on how to better protect my digital Asset Here's Their Website Info www cyberconstableintelligence com
15.02.25 02:51 Michelle Lynn

Living in Los Angeles, I never imagined I’d face such a difficult chapter in my life. At the time, my wife was pregnant, and we were both excited about starting a family. I fell victim to a series of scams, losing over $170,000 in total. Just when I thought things couldn’t get worse, I received a call from someone who promised to help me recover my losses. Desperate to fix the situation, I went along with it, hoping for a breakthrough. But it turned out to be another scam. However, most of the options I found either seemed dubious or offered no real guarantees. That’s when I came across Cyber Constable Intelligence. It was a company recommended in a Facebook community The team worked tirelessly on my case, and after some time, they successfully recovered 99% of my investment. Although I didn’t recover everything, the 99% recovery was a huge relief They also educated me on how to better protect my digital Asset Here's Their Website Info www cyberconstableintelligence com, WhatsApp Info: 1 (252) 378-7611
16.02.25 01:01 Peter

I fell victim to a crypto scam and lost a significant amount of money. What are the most effective strategies to recover my funds? I've heard about legal actions, contacting authorities, and hiring recovery experts, but I'm not sure where to start. Can you provide some guidance on the best ways to recover money lost in a crypto scam? Well if this is you, [email protected] gat you covered get in touch and thank me later
16.02.25 20:06 eunice49954

Agent Lopez specializes in recovering stolen cryptocurrencies, especially Bitcoin/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
18.02.25 19:35 donovancristina

Now, I’m that person sharing my success story on LinkedIn, telling others about the amazing team at TECH CYBER FORCE RECOVERY who literally saved my financial life. I’ve also become that guy who proudly shares advice like “Always back up your wallet, and if you don’t have TECH CYBER FORCE RECOVERY on speed dial.” So, a big thank you to TECH CYBER FORCE RECOVERY if I ever get a chance to meet the team, I might just offer to buy them a drink. They’ve earned it. FOR CRYPTO HIRING WEBSITE WWW://techcyberforcerecovery.com WHATSAPP : ⏩ wa.me/15617263697
18.02.25 22:13 keithphillip671

WhatsApp +44,7,4,9,3,5,1,3,3,8,5 Telegram @Franciscohack The day my son uncovered the truth—that the man I entrusted my hopes of wealth and companionship with through a cryptocurrency platform was a cunning scammer was the day my world crumbled. The staggering realization that I had been swindled out of 150,000.00 Euro worth of Bitcoin left me in a state of profound despair. As a 73-year-old grappling with loneliness, I had sought solace in what I believed to be a genuine connection, only to find deceit and betrayal. Countless sleepless nights were spent in tears, mourning not only the financial devastation but also the crushing blow to my trust. Attempts to verify the authenticity of our interactions were met with hostility, further deepening my sense of isolation. Through the loss it was my son who became my beacon of resilience. He took upon himself the arduous task of tracing the scam and seeking justice on my behalf. Through meticulous effort and determination, he unearthed {F R A N C I S C O H A C K}, renowned for their expertise in recovering funds lost to cryptocurrency scams. Entrusting them with screenshots and evidence of the fraudulent transactions, my son initiated the journey to reclaim what had been callously taken from me. {F R A N C I S C O H A C K} approached our plight with empathy and unwavering professionalism, immediately instilling a sense of confidence in their abilities. Despite my initial skepticism, their transparent communication and methodical approach reassured us throughout the recovery process. Regular updates on their progress and insights into their strategies provided much-needed reassurance and kept our hopes alive amid the uncertainty. Their commitment to transparency and client welfare was evident in every interaction, fostering a sense of partnership rather than mere service. Miraculously, in what felt like an eternity but was actually an impressively brief period, {F R A N C I S C O H A C K} delivered the astonishing news—I had recovered the entire 150,000.00 Euro worth of stolen Bitcoin. The flood of relief and disbelief was overwhelming, marking not just the restitution of financial losses but the restoration of my faith in justice. {F R A N C I S C O H A C K} proficiency in navigating the intricate landscape of blockchain technology and online fraud was nothing short of extraordinary. Their dedication to securing justice and restoring client confidence set them apart as more than just experts—they were steadfast allies in a fight against digital deceit. What resonated deeply with me {F R A N C I S C O H A C K} integrity and compassion. Despite the monumental recovery, they maintained transparency regarding their fees and ensured fairness in all dealings. Their proactive guidance on cybersecurity measures further underscored their commitment to safeguarding clients from future threats. It was clear that their mission extended beyond recovery—it encompassed education, prevention, and genuine advocacy for those ensnared by cyber fraud. ([email protected]) fills me with profound gratitude. They not only rescued my financial security but also provided invaluable emotional support during a time of profound vulnerability. To anyone navigating the aftermath of cryptocurrency fraud, I wholeheartedly endorse {F R A N C I S C O H A C K}. They epitomize integrity, expertise, and unwavering dedication to their clients' well-being. My experience with {F R A N C I S C O H A C K} transcended mere recovery—it was a transformative journey of resilience, restoration, and renewed hope in the face of adversity.
21.02.25 07:42 daniel231101

I never thought I would fall victim to a crypto scam until I was convinced of a crypto investment scam that saw me lose all my entire assets worth $487,000 to a crypto investment manager who convinced me I could earn more from my investment. I thought it was all gone for good but I kept looking for ways to get back my stolen crypto assets and finally came across Ethical Hack Recovery, a crypto recovery/spying company that has been very successful in the recovery of crypto for many other victims of crypto scams and people who lost access to their crypto. I’m truly grateful for their help as I was able to recover my stolen crypto assets and get my life back together. I highly recommend their services EMAIL ETHICALHACKERS009 AT @GMAIL DOT COM whatsapp +14106350697
21.02.25 21:38 eunice49954

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
22.02.25 18:01 benluna0991

Mark Zuckerberg. That’s the name I was introduced to when I first encountered the cryptocurrency mining platform, WHATS Invest. A person claiming to be Zuckerberg himself reached out to me, saying that he was personally backing the platform to help investors like me earn passive income. At first, I was skeptical—after all, how often do you get a direct connection to one of the world’s most famous tech entrepreneurs? But this individual seemed convincing and assured me that many people were already seeing substantial returns on their investments. He promised me a great opportunity to secure my financial future, so I decided to take the plunge and invest $10,000 into WHATS Invest. They told me that I could expect to see significant returns in just a few months, with payouts of at least $1,500 or more each month. I was excited, believing this would be my way out of financial struggles. However, as time passed, things didn’t go according to plan. Months went by, and I received very little communication. When I finally did receive a payout, it was nowhere near the $1,500 I was promised. Instead, I received just $200, barely 13% of what I had expected. Frustrated, I contacted the support team, but the responses were vague and unhelpful. No clear answers or solutions were offered, and my trust in the platform quickly started to erode. It became painfully clear that I wasn’t going to get anywhere with WHATS Invest, and I began to worry that my $10,000 might be lost for good. That's when I discovered Certified Recovery Services. Desperate to recover my funds, I decided to reach out to them for help. In just 24 hours, they worked tirelessly to recover the majority of my funds, successfully retrieving $8,500 85% of my initial investment. I couldn’t believe how quickly and efficiently they worked to get my money back. I’m extremely grateful for Certified Recovery Servicer's fast and professional service. Without them, I would have been left with a significant loss, and I would have had no idea how to move forward. If you find yourself in a similar situation with WHATS Invest or any other platform that isn’t delivering as promised, I highly recommend reaching out to Certified Recovery Services They were a lifesaver for me, helping me recover nearly all of my funds. It's reassuring to know that trustworthy services like this exist to help people when things go wrong. They also specialize in recovering money lost to online scams, so if you’ve fallen victim to such a scam, don’t hesitate to contact Certified Recovery Services they can help! Here's Their Info Below: WhatsApp: +1(740)258‑1417 mail: [email protected], [email protected] Website info; https://certifiedrecoveryservices.com
23.02.25 22:00 eunice49954

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 06:36 ANDREW DAVIS

RECOVER YOUR SCAMMED FUNDS AND CRYPTOCURRENCY VIA SPOTLIGHT RECOVERY Professional hackers at Spotlight Recovery provide services for compromised devices, accounts, and websites as well as for recovering stolen bitcoin and money from scams. They finish their work safely and quickly. Their order has been fulfilled since day one, and the victim will never be conscious of the outside entrance. Very few even attempt to give critical information, look into network security, or discreetly discuss personal issues. The Spotlight Recovery Crew helped me recover $264,000 that was stolen from my corporate bitcoin wallet, and I appreciate them giving me further details on the unidentified people. In the event that you have been defrauded of your hard-earned cash or bitcoins, contact SPOTLIGHT RECOVERY CREW at Contact: [email protected]
24.02.25 07:19 maggie4567

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 07:19 maggie4567

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 07:19 maggie4567

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
27.02.25 12:46 monikaguttmacher

Weddings are supposed to be magical, but the months leading up to mine were anything but. Already, wedding planning was a high-stress, sleep-deprived whirlwind: endless details to manage, from venue deposits and guest lists to dress fittings and vendor contracts. But nothing-and I mean, nothing-compared to the panic that washed over me when I realized that somehow, I had lost access to my Bitcoin wallet-with $600,000 inside. It happened in the worst possible way. In between juggling my to-do lists and trying to keep my sanity intact, I lost my seed phrase. I went through my apartment like a tornado, flipping through notebooks, checking every email, every file-nothing. I sat there in stunned silence, heart pounding, trying to process the fact that my entire savings, my security, and my financial future might have just vanished. In utter despair, I vented to my bridesmaid's group chat for some sympathetic words from the girls. Instead, one casually threw out a name that would change everything in a second: "Have you ever heard of Tech Cyber Force Recovery? They recovered Bitcoin for my cousin. You should call them." I had never heard of them before, but at that moment, I would have tried anything. I immediately looked them up, scoured reviews, and found story after story of people just like me—people who thought they had lost everything, only for Tech Cyber Force Recovery to pull off the impossible. That was all the convincing I needed. From the very first call, I knew I was in good hands. Their team was calm, professional, and incredibly knowledgeable. They explained the recovery process in a way that made sense, even through my stress-fogged brain. Every step of the way, they kept me informed, reassured me, and made me feel like this nightmare actually had a solution. And then, just a few days later, I got the message: "We have recovered your Bitcoin." (EMAIL. support @ tech cyber force recovery . com) OR WHATSAPP (+1 56 17 26 36 97) I could hardly believe my eyes: Six. Hundred. Thousand. Dollars. In my hands again. I let out my longest breath ever and almost cried, relieved. It felt like I woke up from a bad dream, but it was real, and Tech Cyber Force Recovery had done it. Because of them, I walked down the aisle not just as a bride, but as someone who had dodged financial catastrophe. Instead of spending my honeymoon stressing over lost funds, I got to actually enjoy it—knowing that my wallet, and my future, were secure. Would I refer to them? In a heartbeat. If you ever find yourself in that situation, please don't freak out, just call Tech Cyber Force Recovery. They really are the real deal.
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07.03.25 15:18 Ariduk

BEST LINK FOR RECOVERY SCAM ON TRADING INVESTMENT AND OTHERS TROUBLESHOOT? Welcome To General Hacking Techniques Service known as DHACKERS. Year 2025 we are active and best in what we do, as we give Solution to every problem concerning Web3 INTERNET activities, we guide you right to a positive fund Recovery e.t.c. Question From Most of Our Client, HOW POSSIBLE AND TIME WILL IT TAKE TO RECOVERY LOST OR SCAM FUND? Our Answer. Yes is 89.9% possible and how long it takes your Fund to be recovered depend on you. The fact is there are lot of fake binary investment companies out their, same a lot fake recovery companies and agents too. CAUTION 1). Make sure you ask one or two questions concerning the service and how they render there recovery services. 2) Do not give out your scammed details to any agent or hacker when you are not yet ready to recover back your fund. 3) do not make any payment when you are not sure of the service if you are to make one. (4) We discovered that most of this fake hacker or agent do give us scam details, playing the victim, because they can afford the service charge. 5) As long you have your scam details with you, you can recover your fund back anytime any-day with the right channel. Contact us from our Front Desk. [email protected] For advice and services, our standby Guru email. [email protected] [email protected] List of Service. ▶️Binary Recovery ▶️Data Recovery ▶️University Result Upgraded ▶️Clear your Internet Blunder and controversy ➡️Increase your Credit Score ➡️Wiping of Criminal Records ➡️Social Media Hack ➡️Blank ATM Card ➡️Load and wipe ➡️Phone Hacking ➡️Private Key Reset etc. For quick response. Email [email protected] Border us with your jobs and allow us give you positive result with our hacking skills. All Right Reserved (c)2025
08.03.25 04:42 andreassenhedda

If you’ve fallen victim to online scammers who have deceived you into investing your hard-earned money through fraudulent Bitcoin schemes, know that you're not alone. Countless individuals have been misled by scammers using various tactics to swindle money through deceptive investment platforms or promises of high returns. These scams often come in the form of fake cryptocurrency investments, Ponzi schemes, or phishing scams designed to steal your Bitcoin and other assets. Unfortunately, many victims suffer not only financial loss but also the psychological toll of realizing they’ve been taken advantage of. In some cases, scammers go even further, threatening legal consequences or using intimidation to keep victims silent. However, all hope is not lost. There are ways to recover your funds and potentially even bring those responsible to justice. One promising resource that can assist in reclaiming lost funds is Lee Ultimate Hacker, a trusted platform designed to help victims of online fraud. This service specializes in helping individuals who have been scammed through cryptocurrency investments or similar schemes. The platform’s expertise can help you navigate the process of recovering your money, giving you a chance to fight back against those who’ve wronged you. To begin the recovery process, it’s important to gather any proof of payment or transaction history involving the scammers. This evidence is crucial in tracing the flow of funds and identifying the scam operation behind it. Once you have collected your documentation, you can reach out to Lee Ultimate Hacker via LEEULTIMATEHACKER @ AOL . COM or wh@tsapp +1 (715) 314 - 9248 for assistance. They offer professional services to investigate fraudulent schemes, track Bitcoin transactions, and work to reverse fraudulent transfers. The recovery process can be complex and requires expert knowledge of blockchain technology and financial investigations, but with the help of a dedicated team, you’ll have a better chance of seeing your funds returned. In addition to helping you recover your assets, Lee Ultimate Hacker also works towards identifying the scammers and reporting them to the appropriate authorities. They understand the urgency of halting these fraudsters before they can victimize others. With their assistance, you not only increase the chances of recovering your funds but also play a part in holding cybercriminals accountable. If you've been affected by Bitcoin scams or other fraudulent online activities, reaching out to a service like Lee Ultimate Hacker can provide hope and a clear path forward. Their team can guide you through the recovery process, offering expert support while you take steps to reclaim what you’ve lost. It’s time to take action and work toward reclaiming your hard-earned money.
08.03.25 18:20 Diegolola514gmail.com

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08.03.25 18:23 faridasumadi

WEBSITE W.W.W.techcyberforcerecovery.com WHATSAPP +1 561.726.36.97 EMAIL [email protected] I Thought I Was Too Smart to Be Scammed, Until I Was. I'm an attorney, so precision and caution are second nature to me. My life is one of airtight contracts and triple-checking every single detail. I'm the one people come to for counsel. But none of that counted for anything on the day I lost $750,000 in Bitcoin to a scam. It started with what seemed like a normal email, polished, professional, with the same logo as my cryptocurrency exchange's support team. I was between client meetings, juggling calls and drafting agreements, when it arrived. The email warned of "suspicious activity" on my account. My heart pounding, I reacted reflexively. I clicked on the link. I entered my login credentials. I verified my wallet address. The reality hit me like a blow to the chest. My balance was zero seconds later. The screen went dim as horror roiled in my stomach. The Bitcoin I had worked so hard to accumulate over the years, stored for my retirement and my children's future, was gone. I felt embarrassed. Lawyers are supposed to outwit criminals, not get preyed on by them. Mortified, I asked a client, a cybersecurity specialist, for advice, expecting criticism. But he just suggested TECH CYBER FORCE RECOVERY. He assured me that they dealt with delicate situations like mine. I was confident from the first call that I was in good hands. They treated me with empathy and discretion by their staff, no patronizing lectures. They understood the sensitive nature of my business and assured me of complete confidentiality. Their forensic experts dove into blockchain analysis with attention to detail that rivaled my own legal work. They tracked the stolen money through a complex network of offshore wallets and cryptocurrency tumblers tech jargon that appeared right out of a spy thriller. Once they had identified the thieves, they initiated a blockchain reversal process, a cutting-edge method I was not even aware was possible. Three weeks of suffering later, my Bitcoin was back. Every Satoshi counted for. I sat in front of my desk, looking at the refilled balance, tears withheld. TECH CYBER FORCE RECOVERY not only restored my assets, they provided legal-grade documentation that empowered me to bring charges against the scammers. Today, I share my story with colleagues as a warning. Even the best minds get it. But when they do, it is nice to know the Wizards have your back.
09.03.25 17:22 Wayne707

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