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Правда ли KAN лучше MLP? Свойство разделения глубины между двумя архитектурами

Введение

Прошлым летом в свет вышла новая архитектура нейронных сетей под названием Kolmogorov-Arnold Networks (KAN). Основная статья есть в открытом доступе на архиве по следующей ссылке. На момент выхода статьи эта новость произвела фурор в мире машинного обучение, так как KAN показывали существенный прирост в качестве аппроксимации различных сложных функций. На фото ниже видно, что ошибка новых сетей падает значительно быстрее при увеличении числа параметров.

графики RMSE на тестовой выборке к числу параметров у KAN и MLP на разных функциях

Однако, за все приходится платить, и цена таких маленьких значений функции ошибки - медленное обучение: KAN обучается примерно в 10 раз медленнее, чем старый добрый MLP. Из всего этого возникает вопрос: насколько все же уместно использование новой архитектуры вместо привычных всем MLP?

Конечно же, ответ на этот вопрос не столь однозначный: кому-то больше важна скорость обучения и работы модели, кому-то качество предсказаний. В поисках ответа я пришел к интересной задаче, которая легла в основу моей курсовой работы (полная версия по ссылке) и которую я собираюсь здесь изложить.

Разделение глубины и постановка задачи

Один из способов сравнить архитектуры - выявить так называемое свойство разделения глубины. Для MLP сетей оно заключается в следующем: пусть есть MLP сеть $\mathcal{N}_1$ глубины l_1 и MLP сеть $\mathcal{N}_{2}$ глубины l_2 , при этом l_1 < l_2 , а ширина обоих сетей полиномиальна относительно размерности входного пространства (далее обозначаем через $\mathcal{poly}(d)$ ) если существует функция такая, что она приближается (в каком именно смысле будет понятно позднее) сетью $\mathcal{N}_1$ , но не приближается $\mathcal{N}_{2}$ , то говорят, что выполняется свойство разделения глубины между $\mathcal{N}_1$ и $\mathcal{N}_{2}$ .

Моя идея заключалась в том, чтобы попытаться установить подобное свойство, которое разделяло бы KAN и MLP. Конкретно, вопрос ставился следующим образом: пусть $\mathcal{K}$ - KAN глубины , ширины $\mathcal{poly}(d)$ и $\mathcal{N}$ - MLP сеть той же глубины и полиномиальной ширины. Существует ли такая функция , что она приближается $\mathcal{K}$ , но не приближается $\mathcal{N}$ ?

Для того, чтобы ответить на этот вопрос, нам придется вникнуть в устройство KAN, рассмотреть один из известных результатов разделения глубины для MLP сетей глубины 2 и 3 (ширины $\mathcal{poly}(d)$ ), а также объединить это все с помощью результата по преобразованию кусочно-линейного (станет понятно далее) KAN в MLP и обратно. Итак, начнем по порядку.

1. Устройство KAN

Для начала определим, что такое слой KAN.

Определение: Слой сети КАN (KAN-слой) с входной размерностью $n_{\text{in}}$ и выходной размерностью $n_{\text{out}}$ задаётся матрицей $\mathbf{\Phi} = \{\phi^{q}_{p}\}_{p=1,\ldots,n_{\text{in}}}^{q=1,\ldots,n_{\text{out}}}$ вещественных функций одной переменной $\phi^{q}_{p}\colon \mathbb{R} \to \mathbb{R}$ , которые мы называем функциями активации. Он представляет функцию

$\mathbf{\Phi}\colon\mathbb{R}^{n_{\text{in}}}\to\mathbb{R}^{n_{\text{out}}}$ $\mathbf{x} = (x_i)_{i=1,\ldots,n_{\text{in}}} \mapsto \mathbf{\Phi}\mathbf(x) = \left(\sum_{i=1}^{n_{\text{in}}}\phi^{j}_{i}(x_i)\right)_{j=1,\ldots,n_{\text{out}}}$

На фото ниже представлен в схематичном виде слой KAN (для удобства связи проведены только между входом и первым выходом, т.е $y_1 = \phi^{1}_{1}(x_1) + \phi^{1}_{2}(x_2) + \dots + \phi^{1}_{n_{in}}(x_{n_{in}})$ ).

Схематичный слой KAN

Теперь определим KAN как композицию слоев.

Определение: Сеть Колмогорова-Арнольда (KAN) глубины — это композиция KAN-слоёв $\mathbf{\Phi}_{L-1} \circ \ldots \circ \mathbf{\Phi}_0$ .

Например, в случае, когда последний слой имеет выходную размерность 1, функция, задаваемая KAN, будет иметь такой страшный вид:

$f(\mathbf{x}) = \sum_{i_{L-1}=1}^{n_{L-1}}\phi^{L-1,i_{L}}_{i_{L-1}}\!\left(\ldots \sum_{i_1=1}^{n_1}\phi^{1,i_2}_{i_1}\left(\sum_{i_0=1}^{n_0}\phi^{0,i_1}_{i_0}(x_{i_0})\right)\ldots\right)$

где $n_\ell$ — входная размерность $\ell$ -го KAN-слоя $\mathbf{\Phi}_\ell = \{\phi^{\ell,q}_{p}\}_{p=1,\ldots,n_\ell}^{q=1,\ldots,n_{\ell+1}}$ .

Добавлю также пару слов про механизм обучения KAN. Изначально все $\phi^{\ell,q}_{p}$ задаются в виде сплайнов и некоторой нелинейной добавки b(x) , т.е $\phi^{\ell,q}_{p} = w_bb_p^{l,q}(x) + w_sspline_p^{l,q}(x)$ где

$spline_p^{l,q}(x) = \sum_{i}{c_{i}B_i(x)}$

есть линейная комбинация B-сплайнов. Для тех, кто не знаком со сплайнами, добавлю: сплайн - это просто кусочно-полиномиальная функция (многочлен фиксированной степени на каждом отрезке разбиения), а B-сплайны - базис в пространстве таких функций. На фото ниже можно посмотреть пример интерполяции функции сплайнами разной степени и B-сплайнами.

Пример интерполяции сплайнами

Сейчас и далее не столь важно, как строятся эти самые сплайны, мы хотим понять основной механизм обучения. При инициализации w_s = 1 , $spline(x) \approx 0$ и b(x) определяется в соответствии с инициализацией Xavier. Далее происходит процесс обратного распространения ошибки: вычисляется градиент функции потерь по параметрам модели, в которые входят коэффициенты c_i в реализациях сплайнов (обычно берутся кубические сплайны и итоговые функции являются гладкими), а также веса w_b и w_s , затем делается шаг в направлении антиградиента и процесс повторяется снова. При обучении происходит так называемое расширение сетки: каждые k шагов обучения мы увеличиваем число точек разбиения отрезков для аппроксимации сплайнами, тем самым увеличивая число параметров c_i сети.

2. От KAN к MLP и обратно

Для начала введем несколько определений, которые понадобятся в этом разделе:

Определение: Кусочно-линейный KAN - это KAN, в котором все функции $\phi^{q}_{p}$ - кусочно-линейные.

Определение: Через $\mathbf{KAN}{(L, n, k)}$ будем обозначать семейство функций, задаваемых с помощью кусочно-линейного KAN глубины и ширины . Здесь - число сегментов кусочно-линейных функций $\phi^{q}_{p}\colon \mathbb{R} \to \mathbb{R}$ .

Определение: Через $\mathbf{ReLU}{(L, n)}$ будем обозначать семейство функций, задаваемых с помощью ReLU MLP сети глубины и ширины .

В попытках найти подход к изначальной задаче о разделении глубины я наткнулся на статью Relating Piecewise Linear Kolmogorov Arnold Networks to ReLU Networks, в которой показано, что кусочно-линейный KAN ширины и глубины может быть представлен ReLU MLP сетью глубины на 1 больше, ширины n^2(k+1) . Более строго, верна следующая теорема:

Теорема: Для любого кусочно-линейного KAN $f\colon \mathbb{R}^n \to \mathbb{R}$ существует ReLU-сеть $g\colon \mathbb{R}^n \to \mathbb{R}$ такая, что f(x) = g(x) для всех $x \in \mathbb{R}^n$ , при этом после перехода к MLP глубина увеличивается на 1, а ширина становится равной n^2(k+1) , где k - число сегментов кусочно-линейных функций $\phi^{q}_{p}$ . (верно и для $f\colon \mathbb{R}^n \to \mathbb{R}^m$ ).

В этой же статье доказывается и обратный факт:

Теорема: Пусть $g\colon \mathbb{R}^n \to \mathbb{R}^m$ — это MLP-сеть с функциями активации из семейства $\mathcal{F}$ . Тогда существует KAN $f\colon \mathbb{R}^n \to \mathbb{R}^m$ с функциями активации, которые являются либо аффинно-линейными, либо принадлежат $\mathcal{F}$ , такой что f(x) = g(x) для всех $x \in \mathbb{R}^n$ , при этом глубина и ширина сети не меняются при переходе к KAN. В частности, если — это ReLU-сеть, то существует кусочно-линейный KAN , для которого f(x) = g(x) при всех $x \in \mathbb{R}^n$ .

Из этих теорема мы можем понять, что

$\mathbf{KAN}(L, n, k) \subseteq \mathbf{ReLU}(L+1, n^2(k+1)) \subseteq \mathbf{KAN}(L+1,n^2(k+1),1)$

В частности

$\mathbf{KAN}(2, poly(d), k) \subseteq \mathbf{ReLU}{(3, poly(d))}$

Иначе говоря, любой двухслойный кусочно-линейный KAN ширины poly(d) может быть реализован в виде трехслойной ReLU MLP сети, имеющей полиномиальную ширину (на всякий случай нужно уточнить, что ширина у них не обязана совпадать, она просто должна быть полиномиальной относительно входной размеренности).
Таким образом, интерес представляет следующий вопрос: если мы можем представить любой двухслойный кусочно-линейный KAN трехслойной ReLU MLP, может мы можем представить его ограничившись лишь двумя слоями?
Отрицательный ответ на этот вопрос вел бы к отсутствию вложения

$\mathbf{KAN}{(2, poly(d), k)} \not\subseteq \mathbf{ReLU}{(2, poly(d))}$

а следовательно было бы установлено свойство разделения глубины между двухслойным кусочно-линейным (а значит и произвольным) KAN и двухслойной ReLU MLP (везде ширина предполагается poly(d) ). То есть, существовала бы такая функция , что она реализуется двухслойным KAN, но не реализуется двухслойной ReLU MLP.
На самом деле, мы можем пойти наоборот: построим функцию , а затем покажем, что она удовлетворяет нужным условиям. Как раз этому посвящены следующие разделы.

3. Поиск искомой функции

В этом разделе нам понадобятся 2 предположения об используемых активационных функциях MLP-сетей:

Предположение 1: Для заданной функции активации $\sigma$ существует константа $c_{\sigma} \geq 1$ (зависящая только от $\sigma$ ), такая что выполняется следующее:
Для любой -липшицевой функции $f\colon \mathbb{R} \rightarrow \mathbb{R}$ , которая постоянна вне ограниченного интервала [-R, R] , и для любого $\delta \gt 0$ существуют скаляры $a, \{\alpha_i, \beta_i, \gamma_i\}_{i=1}^w$ , где $w \leq c_{\sigma}\frac{RL}{\delta}$ , такие что функция

$h(x) = a + \sum_{i=1}^w \alpha_i \cdot \sigma(\beta_i x - \gamma_i)$

удовлетворяет условию:

$\sup_{x \in \mathbb{R}} |f(x) - h(x)| \leq \delta.$

Предположение 2: Функция активации $\sigma$ является (лебегово) измеримой и удовлетворяет условию:

$|\sigma(x)| \leq C(1 + |x|^\alpha)$

для всех $x \in \mathbb{R}$ и некоторых констант $C, \alpha \gt 0$ .

Стоит заметить, что эти два предположения выполняются для многих известных активационных функций, в частности для ReLU. Первое предположение говорит о том, что мы можем приблизить любую липшицеву функцию с ограниченным носителем с помощью двухслойной MLP сети с данной активационной функцией, а второе ограничивает рост активационной функции некоторым полиномом.

В процессе поиска подходящего кандидата на роль "плохой" функции я вспомнил про статью 2016 года с результатом разделения глубины для MLP сетей глубины 2 и 3 - The Power of Depth for Feedforward Neural Networks. Основным результатом данной работы является следующая теорема:

Теорема: Предположим, что функция активации $\sigma(\cdot)$ удовлетворяет предположению 1 с константой $c_{\sigma}$ , а также предположению 2. Тогда существуют универсальные константы $c, C \gt 0$ такие, что для любой размерности $d \gt C$ найдется вероятностная мера $\mu$ на $\mathbb{R}^{d}$ и функция $\tilde{g}: \mathbb{R}^{d} \to \mathbb{R}$ со следующими свойствами:

Функция $\tilde{g}$ ограничена в , имеет носитель на $\{\mathbf{x} : \|\mathbf{x}\| \leq C\sqrt{d}\}$ и может быть представлена 3-слойной нейронной сетью с шириной не более $Cc_{\sigma}d^{19/4}$ .
Любая функция , представленная двухслойной сетью с шириной не более $ce^{cd}$ , удовлетворяет неравенству:

$\mathbb{E}_{\mathbf{x}\sim\mu}\left(f(\mathbf{x}) - \tilde{g}(\mathbf{x})\right)^2 \geq c.$

То есть, теорема говорит о том, что существует некоторая функция $\tilde{g}$ и вероятностная мера $\mu$ , что $\tilde{g}$ не приближается никакой двухслойной ReLU MLP сетью полиномиальной ширины по норме $L_2{(\mu)}$ , но трехслойной сети уже достаточно для ее приближения.

Моя идея заключалась в том, что такая функция $\tilde{g}$ может подойти и для моей задачи. Для этого надо было проверить, возможно ли приблизить ее с помощью двухслойного кусочно-линейного KAN.

Перед тем, как доказывать, что $\tilde{g}$ действительно приближается KAN-ом, обсудим ее построение.

Для начала зададим вероятностную меру $\mu$ на $\mathbb{R}^d$ . Это будет такая мера, что ее плотность имеет вид $\phi^2(\mathbf{x})$ , где $\phi$ - преобразование Фурье индикатора единичного евклидова шара $\mathbf{1}\{\mathbf{w} \in R_dB_d\}$ , здесь R_d - множитель, такой, что Vol(R_dB_d) = 1 . График такой плотности для d = 2 приведен ниже, а явная формула имеет вид

$\varphi(\mathbf{x}) = \left(\frac{R_d}{\|\mathbf{x}\|}\right)^{d/2} J_{d/2}(2\pi R_d \|\mathbf{x}\|)$

где $J_{d/2}(\cdot)$ - функция Бесселя первого типа, порядка d/2 . (график для $J_{20}(\cdot)$ ниже)

График искомой плотности, справа - увеличенный масштаб

График функции Бесселя первого типа порядка 20

Идея брать именно такую меру исходит из того, что носитель двухслойной нейронной сети представляет из себя объединение многомерных трубок ограниченного радиуса, проходящих через начало координат. А это значит, что функция $\widehat{\tilde{g}*\phi}$ (преобразование Фурье произведение $\tilde{g}$ и $\phi$ ) должна иметь большую часть своей $L_2{(\mu)}$ -массы далеко от начала координат. Но $\widehat{\tilde{g}\phi} = \hat{\tilde{g}} \ast \hat{\phi}$ (справа - свертка преобразований Фурье), а свертка с индикатором единичного шара как раз "разносит" массу от центра (это известный факт для L_1 , но из построения это будет верно и для L_2 ). Подробнее об этом можно прочитать в исходной статье в разделе "Proof Sketch".

Теперь зададим $\tilde{g}$ как комбинацию индикаторов некоторых "тонких" оболочек:

$\tilde{g}(\mathbf{x}) = \sum_{i=1}^N \epsilon_i g_i(\mathbf{x})$

где $\epsilon_i \in \{-1, +1\}$ , N = poly(d) и $g_i(\mathbf{x}) = \mathbf{1}\{\|\mathbf{x}\| \in \Delta_i\}$ , где $\Delta_i$ - непересекающиеся интервалы ширины $\mathcal{O}{(1/N)}$ , значения которых лежат в пределах $\mathcal{\Theta}{(\sqrt{d})}$ . Более подробное построение можно посмотреть в исходной статье, либо в полном тексте моей курсовой.

4. Приближение с помощью KAN

Теперь нам остается проверить, что построенная в предыдущем разделе функция действительно приближается двухслойным кусочно-линейным KAN. Тогда, в силу того, что двухслойной ReLU MLP она не приближается, будет получена невложимость классов $\mathbf{KAN}{(2, poly(d), k)} \not\subseteq \mathbf{ReLU}{(2, poly(d))}$ .

Начнем с переформулировки Предположения 1 из предыдущего раздела для архитектуры KAN:

Предположение 1(KAN): Пусть $\phi$ - активационная функция (сплайн с K + 1 сегментами), тогда для любой -липшицевой функции $f: \mathbb{R} \to \mathbb{R}$ , такой, что $f \equiv const$ вне интервала [-R, R] и для любого $\delta \gt 0$ найдется $K \in \mathbb{N}$ , такое, что верно:

$\sup_{x\in\mathbb{R}}|f(x) - \phi(x)| \leq \delta\$

Заметим, что для кусочно-линейных сплайнов это предположение выполняется. Отрезок [-R, R] можно разбить на K равных частей с шагом $h = \frac{2R}{K}$ , затем строим сплайн S(x) , совпадающий с f(x) в точках x_i = -R + ih :

$S(x) = \left\{ \begin{array}{l} f(x_i) + \frac{f(x_{i+1}) - f(x_i)}{h}(x-x_i) \quad x\in[x_i, x_{i+1}], \\ f(x) \equiv const \quad x \notin [-R, R] \\ \end{array} \right.$

Оценим точность приближения, так как f(x) является -липшицевой, то на каждом отрезке $[x_{i}, x_{i+1}]$ :

$\sup_{x\in\mathbb{R}}|f(x)-S(x)| \leq L \cdot h\$

Чтобы выполнялось $L \cdot h \leq \delta$ , достаточно положить $h \leq \frac{\delta}{L}$ , т.е $K \geq \frac{2LR}{\delta}$ .

Также нам понадобится следующая лемма из оригинальной статьи:

Лемма 1: Предположим, что $d \geq 2$ . Для любого выбора $\epsilon_i \in \{-1, +1\}$ , $i = 1, \ldots, N$ существует -липшицева функция , носитель которой содержится в $[\alpha \sqrt{d}, 2\alpha \sqrt{d}]$ и принимающая значения в [-1, +1] , удовлетворяющая условию

$\int\left( f(\mathbf{x}) - \sum_{i=1}^N \epsilon_i g_i(\mathbf{x}) \right)^{2} \varphi^2(\mathbf{x}) dx \leq \frac{3}{\alpha^2 \sqrt{d}}$

Здесь $\alpha \geq 1$ - большая числовая константа, которая должна быть больше определенного порога, также она участвует в построении $\tilde{g}$ (подробнее в оригинальной статье).

Из данной леммы уже видно, что наша "плохая" функция может быть приближена по $L_2{(\mu)}$ -норме некоторой липшицевой функцией многих переменных. Заметим также, что значения $\tilde{g}$ зависят лишь от нормы $\|\mathbf{x}\|$ . Осталось понять, можем ли мы приближать двухслойным кусочно-линейным KAN такие липшицевы функции, зависящий от нормы входного вектора. Об этом следующая лемма:

Лемма 2: Предположим, что функция активации $\phi$ удовлетворяет предположению 1(KAN). Пусть - -липшицева функция с носителем на [r, R] , где $r \geq 1$ . Тогда для любого $\delta \gt 0$ существует функция , представимая двухслойным кусочно-линейным KAN ширины , в котором число сегментов кусочно-линейных функций $K \geq \frac{2RL}{\sqrt{r}\delta}\cdot\max(R, L)$ , такая что

$\sup_{\mathbf{x}\in\mathbb{R}^{d}}|g(\mathbf{x}) - f(\|\mathbf{x}\|)| < \delta.$

Доказательство: Определим -липшицеву функцию

$l(x) = \min(x^2, R^2)$

которая константа вне отрезка [-R, R] . Такими же свойствами обладает и функция

$l(\mathbf{x}) = \sum_{i=1}^{d}l(x_i)$

определенная на $\mathbb{R}^d$ . Применим Предположение 1(KAN) к функции $l(\cdot)$ , получим функцию $\tilde{l}(x) = \phi(x)$ такую, что

$\sup_{x\in\mathbb{R}}|\tilde{l}(x) - l(x)| \leq \frac{\sqrt{r}\delta}{dL}$

Такое неравенство верно при $K \geq \frac{2RL^2}{\sqrt{r}\delta}$ . Следовательно, функция

$\tilde{l}(\mathbf{x}) = \sum_{i=1}^d\tilde{l}(x_i)$

будет иметь вид $\tilde{l}(\mathbf{x}) = \sum_{i=1}^d\phi_i(x_i)$ .

Введем теперь следующую функцию:

$s(x) = \left\{ \begin{array}{l} f(\sqrt{x}) \quad x \geq 0, \\ 0 \quad x < 0 \\ \end{array} \right.$

Так как является -липшицевой с носителем в множестве $\{x \colon r \leq x \leq R\}$ , отсюда следует, что является $\frac{L}{2\sqrt{r}}$ -липшицевой с носителем в [-R^2, R^2] . Снова применим предположение 1(KAN) к функции $s{(\cdot)}$ с $K \geq \frac{2R^2L}{\sqrt{r}\delta}$ . Получим функцию $\tilde{s}(x) = \eta(x)$ , что

$\sup_{x\in\mathbb{R}}|\tilde{s}(x) - s(x)| \leq \frac{\delta}{2}$

Теперь рассмотрим композицию $g = \tilde{s} \circ \tilde{l}$ , которая из определений $\tilde{s}$ и $\tilde{l}$ имеет вид:

$g(x) = \eta\left(\sum_{i=1}^d\phi_i(x_i)\right)$

Т.е представляет собой двухслойный KAN ширины .

Покажем, что $\sup_{\mathbf{x}\in\mathbb{R}^d}|g(\mathbf{x}) - f(\|\mathbf{x}\|)| \leq \delta$ . Для любого $\mathbf{x}$ из $\mathbb{R}^d$ мы имеем:

$|g(\mathbf{x}) - f(\|\mathbf{x}\|)| \leq |\tilde{s}(\tilde{l}(\mathbf{x})) - s(\tilde{l}(\mathbf{x}))| + |s(\tilde{l}(\mathbf{x})) - s(l(\mathbf{x}))| + |s(l(\mathbf{x})) - f(\|\mathbf{x}\|)| =$ $= \\|\tilde{s}(\tilde{l}(\mathbf{x})) - s(\tilde{l}(\mathbf{x}))| + |s(\tilde{l}(\mathbf{x})) - s(l(\mathbf{x}))| + |f(\sqrt{l(\mathbf{x})}) - f(\|\mathbf{x}\|)|$

Рассмотрим каждое из трёх абсолютных значений:

Первое абсолютное значение не превосходит $\delta/2$ согласно построению $\tilde{s}$ .
Второе абсолютное значение, поскольку является $\frac{L}{2\sqrt{r}}$ -липшицевой, не превосходит $\frac{L}{2\sqrt{r}}|\tilde{\ell}(\mathbf{x}) - \ell(\mathbf{x})|$ , что по построению $\tilde{l}$ составляет не более $\delta/2$ .
Что касается третьего абсолютного значения: если $\|\mathbf{x}\|^{2} \leq R^{2}$ , то $\ell(\mathbf{x}) = \|\mathbf{x}\|^{2}$ и значение равно нулю. Если $\|\mathbf{x}\|^{2} > R^{2}$ , то легко проверить, что $\ell(\mathbf{x}) \geq R^{2}$ , и поскольку непрерывна и имеет носитель на , следует, что $f(\sqrt{\ell(\mathbf{x})}) = f(\|\mathbf{x}\|) = 0$ , и снова получаем ноль.

Суммируя вышесказанное, получаем $|g(\mathbf{x}) - f(\|\mathbf{x}\|)| \leq \frac{\delta}{2} + \frac{\delta}{2} + 0 = \delta$ , что и требовалось. Лемма 2 доказана.

Наконец, объединим Лемму 1 и Лемму 2 для доказательства того, что функция $\tilde{g}$ приближается двухслойным KAN:

Теорема: Существует универсальная константа $C \gt 0$ такая, что выполняется следующее. Пусть $\delta \in (0, 1)$ . Предположим, что $d \geq C$ и функции g_i построены как указано выше. Для любого выбора $\epsilon_i \in \{-1, +1\}$ , $i = 1, \ldots, N$ существует функция , представимая двухслойным кусочно-линейным KAN ширины , со значениями в [-2, +2] , такая что

$\| g(x) - \sum_{i=1}^N \epsilon_i g_i(\|x\|)\|_{L_2(\mu)} \leq \frac{\sqrt{3}}{\alpha d^{1/4}} + \delta.$

Доказательство: Сначала применим Лемму 2, чтобы получить -липшицеву функцию со значениями в [-1, +1] , удовлетворяющую условию

$\| h{(\mathbf{x})} - \sum_{i=1}^{N} \epsilon_{i} g_i{(\mathbf{x})} \|_{L_2{(\mu)}} = \sqrt{\int_{\mathbb{R}^d}{\left( h{(\mathbf{x})} - \sum{\epsilon_i g_i{(x)}} \right)^2 \varphi^2(x) dx}} \leq \frac{\sqrt{3}}{\alpha d^{1/4}}.$

Затем используем Лемму 2 с параметрами $R = 2\alpha\sqrt{d}$ , $r = \alpha\sqrt{d}$ , L = N ( тогда $K \geq \frac{8\alpha^{3/2}d^{3/4}N}{\delta}$ ) для построения функции , представимой двухслойным кусочно-линейным KAN ширины , такой что $\sup_{x \in \mathbb{R}^d} |g(x) - h(x)| \leq \delta$ . Это означает, что $\|g - h\|_{L_2(\mu)} \leq \delta$ , а также что значения лежат в $[-1 - \delta, 1 + \delta] \subseteq [-2, +2]$ (поскольку $\delta < 1$ ). Объединяя это с равенством выше и используя неравенство треугольника, завершаем доказательство.

Подведение итогов

Проделав такую большую работу, мы показали, что существует функция $\tilde{g}$ такая, что она не может быть приближена двухслойной ReLU MLP ширины poly(d) , но может быть приближена двухслойным кусочно-линейным KAN ширины poly(d) с числом сегментов $K \geq \frac{8\alpha^{3/2}d^{3/4}N}{\delta}$ . Иначе говоря, доказано отсутствие вложения

$\mathbf{KAN}{\left(2, poly(d), \left\lceil \frac{8\alpha^{3/2}d^{3/4}N}{\delta}\right\rceil + 1\right)} \not\subseteq \mathbf{ReLU}{(2, poly(d))}$

Таким образом, существуют функции, которые не могут быть приближены старой архитектурой MLP, однако могут быть реализованы с помощью KAN (даже в случае кусочно-линейных активационных функций у последнего). Конечно же, этот результат в большей степени теоретический, но он показывает, что в новой архитектуре может быть заложена куда большая выразимостная мощность, чем в классической.

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22 июл. 2025 г.

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"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.10.25 21:52 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.10.25 20:17 tyleradams

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17.10.25 20:20 lindseyvonn

Have you gotten yourself involved in a cryptocurrency scam or any scam at all? If yes, know that you are not alone, there are a lot of people in this same situation. I'm a Health Worker and was a victim of a cryptocurrency scam that cost me a lot of money. This happened a few weeks ago, there’s only one solution which is to talk to the right people, if you don’t do this you will end up being really depressed. I was really devastated until went on LinkedIn one evening after my work hours and i saw lots of reviews popped up on my feed about [email protected], I sent an email to the team who came highly recommended - [email protected] I started seeing some hope for myself from the moment I sent them an email. The good part is they made the entire process stress free for me, i literally sat and waited for them to finish and I received what I lost in my wallet
17.10.25 20:22 richardcharles

I would recommend NVIDIA TECH HACKERS to anyone that needs this service. I decided to get into crypto investment and I ended up getting my crypto lost to an investor late last year. The guy who was supposed to be managing my account turned out to be a scammer all along. I invested 56,000 USD and at first, my reading and profit margins were looking good. I started getting worried when I couldn’t make withdrawals and realized that I’ve been scammed. I came across some of the testimonials that people said about NVIDIA TECH HACKERS and how helpful he has been in recovering their funds. I immediately contacted him in his mail at [email protected] so I can get his assistance. One week into the recovery process the funds were traced and recovered back from the scammer. I can't appreciate him enough for his professionalism.
17.10.25 20:23 stevekalfman

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17.10.25 21:42 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.10.25 21:42 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.10.25 21:42 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.10.25 08:39 debby131

Given how swiftly the cryptocurrency market moves, losing USDT may be a terrifying and upsetting experience. Whether you experienced a technical problem, a transaction error, or were the victim of fraud, it is important to understand the potential recovery routes. Marie can assist you in determining the specific actions you can take to attempt to regain your lost USDT when you need guidance and clarification. You can reach her via email at [email protected] and WhatsApp at +1 7127594675.
21.10.25 11:45 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.10.25 11:45 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.10.25 04:48 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
22.10.25 07:36 donnacollier

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22.10.25 11:59 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.10.25 11:59 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.10.25 16:31 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
23.10.25 01:52 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.10.25 01:52 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.10.25 15:47 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
23.10.25 21:43 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.10.25 21:43 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 06:08 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
24.10.25 06:40 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 06:40 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 06:54 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
24.10.25 18:18 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 18:18 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.10.25 00:49 tyleradams

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25.10.25 00:50 stevekalfman

If you ever need hacking services, look no further. I found myself in a difficult situation after losing almost $510,000 USD in bitcoin. I was distraught and had no hope of survival because i lost my life savings to this scam, I had no chance of recovering my investment. Until i came across an article on google about ([email protected]). A real-life recovery expert, everything changed completely after contacting him and explaining my ordeal to him and he quickly stepped in and assisted me in recovering 95% of my lost funds. They guarantee their clients and i got the highest level of happiness their services are highly recommended. so I’m putting this here for anyone who require their services too" Email: [email protected]
25.10.25 05:05 victoriabenny463

A few months ago, I fell for what looked like a legitimate online trading platform. The website was sleek, the support team was responsive, and they even showed me fake profit charts that seemed to grow daily. I started small, and after a few test withdrawals worked, I invested more eventually depositing 2 Bitcoin in hopes of long-term returns. But one morning, when I tried to withdraw, everything changed. The platform suddenly locked my account, claiming there were additional fees to release my funds. The support chat stopped responding, and the website started giving error messages. That’s when I realized I’d been scammed. I was devastated. Two Bitcoins wasn’t pocket change, it was my savings. I filed reports online, but everyone told me it was impossible to recover crypto once it’s sent. Then, through a discussion forum, I came across Dexdert Net Recovery, a team specializing in tracing and recovering stolen digital assets. Skeptical but desperate, I reached out. Their response was immediate and professional. They asked for transaction IDs, wallet addresses, screenshots of my correspondence with the fake platform, and any proof of payment I had. Within days, their analysts traced the stolen Bitcoin through multiple wallets using blockchain forensics, They updated me step-by-step, Working with their recovery experts and blockchain partners, Dexdert Net Recovery successfully froze the funds and verified ownership of my wallet. After 2 days of persistence and verification, I finally saw my 2 Bitcoin restored back into my own wallet. I couldn’t believe it. What stood out most wasn’t just that they recovered my funds, but how transparent and supportive the Dexdert team was throughout the process. They educated me on crypto safety, how to spot fake trading sites, and how to secure my digital assets in the future. Dexdert Net Recovery truly lives up to its name. They gave me back not just my Bitcoin, but my peace of mind. Reach Out Information Below Via: Email: ([email protected]) Telegram: (https://t.me/Dexdertprorecovery) WhatsApp: (+1 (859) 609‑4156)
25.10.25 10:13 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.10.25 10:13 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.10.25 01:03 Christopherbelle

A terrible financial shock hit me hard. I lost $860,000 in a fake crypto scheme. Things looked good until I tried to cash out my earnings. Suddenly, my account vanished into thin air. I told the police many times, but help never showed up. Despair washed over me; I felt totally beaten. Then, I found Sylvester Bryant Intelligence. Right away, they were skilled and honest about the whole thing. They looked deep into my situation. They tracked where the stolen funds went. They fought hard for me every step of the way. To my great surprise, they got my money back. I thought that was impossible. I owe them so much for their hard work and truthfulness. Sylvester Bryant Intelligence restored my peace. If scams took your crypto, contact them now. Contact Info: Yt7cracker@gmail . com WhatsApp: ‪+1 512 577 7957. or +44 7428 662701.
26.10.25 18:09 victoriabenny463

A few months ago, I fell for what looked like a legitimate online trading platform. The website was sleek, the support team was responsive, and they even showed me fake profit charts that seemed to grow daily. I started small, and after a few test withdrawals worked, I invested more eventually depositing 2 Bitcoin in hopes of long-term returns. But one morning, when I tried to withdraw, everything changed. The platform suddenly locked my account, claiming there were additional fees to release my funds. The support chat stopped responding, and the website started giving error messages. That’s when I realized I’d been scammed. I was devastated. Two Bitcoins wasn’t pocket change, it was my savings. I filed reports online, but everyone told me it was impossible to recover crypto once it’s sent. Then, through a discussion forum, I came across Dexdert Net Recovery, a team specializing in tracing and recovering stolen digital assets. Skeptical but desperate, I reached out. Their response was immediate and professional. They asked for transaction IDs, wallet addresses, screenshots of my correspondence with the fake platform, and any proof of payment I had. Within days, their analysts traced the stolen Bitcoin through multiple wallets using blockchain forensics, They updated me step-by-step, Working with their recovery experts and blockchain partners, Dexdert Net Recovery successfully froze the funds and verified ownership of my wallet. After 2 days of persistence and verification, I finally saw my 2 Bitcoin restored back into my own wallet. I couldn’t believe it. What stood out most wasn’t just that they recovered my funds, but how transparent and supportive the Dexdert team was throughout the process. They educated me on crypto safety, how to spot fake trading sites, and how to secure my digital assets in the future. Dexdert Net Recovery truly lives up to its name. They gave me back not just my Bitcoin, but my peace of mind. Reach Out Information Below Via: Email: ([email protected]) Telegram: (https://t.me/Dexdertprorecovery) WhatsApp: (+1 (859) 609‑4156)
27.10.25 11:15 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.10.25 11:15 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.10.25 17:08 raymondgonzales

Three Cheers for Wizard Hilton Cyber Tech, Bitcoin Savior Extraordinaire In the ever-evolving world of cryptocurrency, where volatility and uncertainty reign supreme, one individual has emerged as a beacon of hope and innovation - Wizard Hilton Cyber Tech, the self-proclaimed "Bitcoin Savior Extraordinaire." This enigmatic figure, with his uncanny ability to navigate the complexities of the digital asset landscape, has captivated the attention of crypto enthusiasts and skeptics alike. Wizard Hilton Cyber Tech journey began as a humble programmer, tinkering with the underlying blockchain technology that powers Bitcoin. But it was his visionary approach and unwavering commitment to the cryptocurrency's potential that propelled him into the limelight. Through a series of strategic investments, groundbreaking algorithmic trading strategies, and a knack for anticipating market shifts, Wizard Hilton Cyber Tech has managed to consistently outperform even the savviest of Wall Street traders. Wizard Hilton Cyber Tech ability to turn even the most tumultuous of Bitcoin price swings into opportunities for substantial gains has earned him the moniker "Wizard," a testament to his unparalleled mastery of the digital currency realm. But Wizard Hilton Cyber Tech impact extends far beyond personal wealth accumulation - Wizard Hilton Cyber Tech has tirelessly advocated for the widespread adoption of Bitcoin, working tirelessly to educate the public, lobby policymakers, and collaborate with industry leaders to overcome the barriers that have long hindered the cryptocurrency's mainstream acceptance. With infectious enthusiasm and boundless energy, Wizard Hilton Cyber Tech has become a true champion of the Bitcoin cause, inspiring a new generation of crypto enthusiasts to embrace the transformative power of this revolutionary technology. As the digital currency landscape continues to evolve, Wizard Hilton Cyber Tech name will undoubtedly be etched in the annals of cryptocurrency history as a visionary, a trailblazer, and, above all, the Bitcoin Savior Extraordinaire. To get your stolen bitcoin back, reach out to Wizard Hilton Cyber Tech via: Email : wizardhiltoncybertech ( @ ) gmail (. ) com WhatsApp number +18737715701 Good Day.
27.10.25 17:20 fatimanorth

The Federal Trade Commission said that more than $1 billion had been lost to cryptocurrency scams in 2023. Victims frequently lose everything, including hope, trust, and wealth. When your hard-earned money disappears into digital shadows, you feel trapped. Nowadays, cryptocurrency frauds are very common. You can get help from recovery specialist Marie at [email protected] and via WhatsApp at +1 7127594675.
28.10.25 00:55 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.10.25 00:55 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.10.25 03:43 Christopherbelle

A sudden money crisis struck me. I dropped $82,000 to a phony crypto scam. All seemed fine until I went to pull out my gains. Then, i was unable to get back my hard earned money. I reported it to the police several times. No aid came my way. Deep sadness hit me; I felt crushed. That's when I learned about Sylvester Bryant Intelligence. They acted quick and fair from the start. They checked my case closely. They followed the path of my lost cash. They pushed strong for me at each turn. In shock, they brought my funds back. I had figured it could not happen. I thank them for their effort and honesty. Sylvester Bryant Intelligence gave me calm again. If a scam stole your crypto, reach out to them today. Contact Info: Yt7cracker@gmail . com WhatsApp: +1 512 577 7957 or +44 7428 662701.
29.10.25 10:56 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.10.25 10:56 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
30.10.25 09:49 Christopherbelle

I suffered a huge money blow when $80,000 slipped away in a bogus crypto plot. It all looked great at first. Then I went to pull out my gains, and poof—my account was gone. I kept telling the cops about it, but they did zip. I felt shattered and lost. That's when I stumbled on Sylvester Bryant Intelligence. Things turned around fast. Right away, they showed clear steps, sharp know-how, and real drive to fix it. They tracked my missing cash step by step and pushed hard till they got it back. I swear, I figured it was a lost cause. Their straight-up work and trust brought back my calm. If a scam stole your crypto, reach out to them now. yt7cracker@gmail . com | WhatsApp: +1 512 577 7957 or +44 7428 662701
30.10.25 12:03 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
31.10.25 13:25 Lillian Lizzy

THANKS TO THE SERVICES OF THE HACK ANGELS // FOR HELPING ME RECOVER MY USDT AND BTC I lost almost $698,000 in a bitcoin investment scam a few months ago. I was devastated and depressed, and I didn't know what to do. When I saw a favorable review of THE HACK ANGELS RECOVERY EXPERT. I decided to contact them and voice my concerns. God is so good that I am a living testament to the fact that there are still legitimate recovery hackers out there. I will confidently recommend THE HACK ANGELS RECOVERY EXPERT to everyone I meet. I will suggest them to anyone who falls victim to any kind of online scam by using the information below. web: https://thehackangels.com Mail Box; [email protected] WhatsApp; +1(520)-200,2320 If you're in London, you can even visit them in person at their office located at 45-46 Red Lion Street, London WC1R 4PF, UK.
31.10.25 13:25 Lillian Lizzy

THANKS TO THE SERVICES OF THE HACK ANGELS // FOR HELPING ME RECOVER MY USDT AND BTC I lost almost $698,000 in a bitcoin investment scam a few months ago. I was devastated and depressed, and I didn't know what to do. When I saw a favorable review of THE HACK ANGELS RECOVERY EXPERT. I decided to contact them and voice my concerns. God is so good that I am a living testament to the fact that there are still legitimate recovery hackers out there. I will confidently recommend THE HACK ANGELS RECOVERY EXPERT to everyone I meet. I will suggest them to anyone who falls victim to any kind of online scam by using the information below. web: https://thehackangels.com Mail Box; [email protected] WhatsApp; +1(520)-200,2320 If you're in London, you can even visit them in person at their office located at 45-46 Red Lion Street, London WC1R 4PF, UK.
01.11.25 03:27 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.11.25 03:27 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.11.25 15:47 kattygates

Hack West Credit Repair is truly in the business of helping people understand credit, how credit works and most importantly they truly care that you understand your next steps to securing a better tomorrow for you and your family. I am grateful that the owner West has taken time out of his evening with his family to get me enrolled and he immediately got the ball rolling in helping me get items deleted from my credit reports. In the few months I’ve been with HACK WEST, 15 of the 17 negative items have been deleted and 250 points have been added to my point. If you want great results, I suggest you use [email protected].
01.11.25 15:49 kattygates

I met an individual on an international dating app, Bumpy. After talking for a few months, the individual encouraged me to start investing on crypto asset trading platform, btc01.org. I started trading and believed I was making profit until I tried to withdraw some of my money, I was asked to pay extra for tax which I did and they kept asking then I knew I have been scammed, I had to contact HACKWEST AT WRITEME DOT COM who then helped to recover the money, the total money I invest is $198,000 in bitcoin. The website is no longer operational. If anyone is in a similar issue, you can as well contact HACK WEST. They also fixed my credit report.
02.11.25 10:55 Christopherbelle

I suffered a crushing blow when $45,000 slipped away in a phony crypto scam. It all looked great at first. But when I went to pull out my gains, my account just vanished. I called the cops over and over. Still, no luck. I felt shattered and lost. That’s when I stumbled on Sylvester Bryant Intelligence. Things turned around fast. Right away, they showed real openness, sharp skills, and a drive to fix it. They tracked my missing money step by step. And they pushed hard until they got it back. I honestly didn’t believe it could happen. Their straight-up approach and solid work brought back my calm. If a scam took your crypto, reach out to them now. yt7cracker@gmail . com | WhatsApp: +1 512 577 7957 or +44 7428 662701
02.11.25 10:56 Christopherbelle

I suffered a crushing blow when $45,000 slipped away in a phony crypto scam. It all looked great at first. But when I went to pull out my gains, my account just vanished. I called the cops over and over. Still, no luck. I felt shattered and lost. That’s when I stumbled on Sylvester Bryant Intelligence. Things turned around fast. Right away, they showed real openness, sharp skills, and a drive to fix it. They tracked my missing money step by step. And they pushed hard until they got it back. I honestly didn’t believe it could happen. Their straight-up approach and solid work brought back my calm. If a scam took your crypto, reach out to them now. yt7cracker@gmail . com | WhatsApp: +1 512 577 7957 or +44 7428 662701
02.11.25 15:23 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
02.11.25 15:23 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.11.25 04:59 Christopherbelle

How to Recover Bitcoin Stolen by a Scammer I dropped £75,000 into a fake crypto scheme. Everything seemed perfect right from the start. But when I tried to cash out my profits, the whole account disappeared. I rang the police time and again. They offered no help at all. I felt broken and alone. Then I found Sylvester Bryant Intelligence. My luck changed in no time. They spoke plainly, worked smart, and focused on my case. They followed the trail of my lost cash bit by bit. They fought until they recovered it all. I never thought it was possible. Their honest approach and expertise restored my peace. If a scam has stolen your crypto, contact them now. [email protected] | WhatsApp: +1 512 577 7957 or +44 7428 662701
04.11.25 09:30 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.11.25 09:30 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
05.11.25 02:43 jordan_11

I made a mistake of not researching about their their investment firm(Binary option), it had a lot of promises which made me fall for all their fake promises , after trying to get my withdrawal out from there, I started having a lot of difficulties. Then I contacted their customer support, they only replied a week after and strangely wanted me to make more deposits just to enable my withdrawal, so i started my research of how I could report this treacherous company to the necessary authorities and also maybe i could get back my investment, The only people that were willing to help me out was Treqora Intel, their customer support replied so fast and immediately directed me to their investigative team which took my up my case and started the tracing and reclaim process with swift, after a week i got an email from Treqora Intel stating that a partial amount had been reclaimed, I was so grateful about their help , then I was asked to decide if I wanted a Partial deposit or wanted all my investment reclaimed and sent to me at once, But I was so desperate at the time so I asked that my partial reclaimed ETH be sent to me first , So the deposit took about few minutes before i got it in my wallet and withdrew it. Then my remaining investment took 2 weeks to get back according to their customer support at Treqora Intel. Immediately They got it, I was contacted again Via email that my reclaimed had been fully reclaimed, so grateful till this day to Treqora Intel for taking their time to help me out with what could have been a very depressing moment to me. And yes no upfront payment until the reclaim has been completed and you've been paid fully. Email:support@treqora. com 📱 WhatsApp: ‪‪‪+1 (7 7 3) 9 7 7 - 7 8 7 7‬ ‬‬ 🌐 Website: Treqora. com.
05.11.25 02:44 jordan_11

Crypto Recovery Services _Hire Treqora Intel
05.11.25 02:48 jordan_11

Legitimate Crypto Recovery Companies-CONSULT TREQORA INTEL. After falling victim to a crypt0currency scam group, I lost $35,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to rec0vermy stolen funds and I came across a lot of testimonials online about TREQORA INTEL , an agent who helps in rec0ver of lost bitc0in funds, I contacted TREQORA INTEL , and with their expertise, they successfully traced and rec0vered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and rec0very protocols. They are trusted and very reliable with a 100% successful rate record Rec0very bitc0in, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypt0. Email:support@treqora. com Whats_App: ‪‪‪‪‪+1 (7 7 3 ) 9 7 7 - 7 8 7 7‬ ‬‬ ‬‬, Website: Treqora dot com.
05.11.25 16:20 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
05.11.25 16:21 robertalfred175

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