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Кластеризация в ML: от теоретических основ популярных алгоритмов к их реализации с нуля на Python

Кластеризация — это набор методов без учителя для группировки данных по определённым критериям в так называемые кластеры, что позволяет выявлять сходства и различия между объектами, а также упрощать их анализ и визуализацию. Из-за частичного сходства в постановке задач с классификацией кластеризацию ещё называют unsupervised classification.

В данной статье описан не только принцип работы популярных алгоритмов кластеризации от простых к более продвинутым, но а также представлены их упрощённые реализации с нуля на Python, отражающие основную идею. Помимо этого, в конце каждого раздела указаны дополнительные источники для более глубокого ознакомления.

Ноутбук с данными алгоритмами можно загрузить на Kaggle (eng) и GitHub (rus).

Содержание

Области применения кластеризации и её разновидности
К-средних
Агломеративная кластеризация
Спектральная кластеризация
DBSCAN
Affinity Propagation

Области применения кластеризации и её разновидности

Кластеризация широко применяется в машинном обучении для решения различного спектра задач:

классификация (определение к какому классу относится каждый объект или же выделение новых классов, которые не были известны заранее);
сегментация рынка (разделение потенциальных клиентов на группы по их характеристикам для разработки более эффективных стратегий в маркетинге и продажах);
сегментация изображений (разделение изображения на сегменты или группы пикселей);
кластеризация геоданных (группировка данных по их географическому расположению, например, разделение районов на безопасные и опасные, богатые и бедные, и так далее);
понижение размерности (уменьшение количества признаков путем объединения схожих в один кластер).

Существует множество различных типов кластеризации, которые можно разделить по следующим критериям:

По способу формирования кластеров:
- Разделительные (partitioning) — разбивают данные на заданное число кластеров, минимизируя расстояние внутри кластера и максимизируя расстояние между кластерами (например, K-means).
- Основанные на плотности (density-based) — группируют точки, которые находятся в областях с высокой плотностью и отделяют их от областей с низкой плотностью (например, DBSCAN).
- Основанные на сетке (grid-based) — разбивают пространство на ячейки сетки и анализируют плотность данных в каждой ячейке (например, STING).
- Основанные на модели (model-based) — предполагают, что данные порождены некоторой статистической моделью и пытаются подобрать параметры этой модели (например, смеси Гауссианов).
- Основанные на графах (graph-based) — используют графовое представление данных и разбивают его на подграфы, соответствующие кластерам (например, спектральная кластеризация).
- Основанные на подпространствах (subspace-based) — ищут кластеры в подпространствах признаков, а не во всём пространстве (например, CLIQUE).
- Основанные на ансамбле (ensemble-based) — комбинируют результаты различных алгоритмов кластеризации, чтобы получить более стабильное и надежное разбиение (например, CSPA).
По степени вложенности кластеров:
- Плоские (flat) — разбивают данные на один уровень кластеров, не учитывая их иерархию (например, K-means).
- Иерархические (hierarchical) — разбивают данные на несколько уровней кластеров, учитывая их иерархию. Существуют два основных подхода к иерархической кластеризации: агломеративный (начинается с того, что каждый объект является отдельным кластером, а затем постепенно наиболее близкие кластеры объединяются в более крупные) и дивизивный (начинается с того, что все объекты составляют один кластер, а затем постепенно разделяются на более мелкие кластеры).
По степени пересечения кластеров:
- Исключающие (exclusive) — каждый объект принадлежит только одному кластеру (например, K-means).
- Перекрывающие (overlapping) — каждый объект может принадлежать нескольким кластерам (например, MCOKE).
- Нечёткие (fuzzy) — каждый объект принадлежит каждому кластеру с некоторой степенью принадлежности (например, fuzzy K-means).

Алгоритмы кластеризации в scikit-learn

Далее более подробно будут рассмотрены наиболее популярные алгоритмы кластеризации, представляющие наибольший интерес с практической точки зрения.

K-Means

Кластеризация K-средних представляет собой одну из самых простых реализаций, суть которой заключается в итеративной инициализации центроидов для каждого кластера на основе среднего арифметического расположенных в нём наблюдений, а также их переопределении путём минимизации суммарного квадратичного отклонения от центроидов кластеров.

Схема кластеризации K-средних

Существуют различные вариации алгоритма K-Means, которые модифицируют его шаги или функцию потерь для улучшения производительности, а также применимости к разным типам данных. К самым популярным вариациям относятся следующие:

Lloyd's algorithm — это классический вариант K-Means, который хорошо работает для сферических кластеров с одинаковой плотностью, но может давать плохие результаты для других форм или размеров кластеров.
Elkan algorithm — более быстрый вариант классического K-Means, который использует неравенство треугольника для уменьшения количества вычислений расстояний между объектами и центроидами, что может быть эффективнее на некоторых наборах данных с хорошо определенными кластерами, однако требуется больше памяти из-за выделения дополнительного массива размера (n_samples, n_clusters).
Mini-batch K-Means — модификация классического K-Means, использующая случайные подвыборки данных на каждой итерации для обучения. Хорошо подходит для больших датасетов.
K-Medoids — вариант K-Means, который в качестве центроидов выбирает реальные точки (медоиды) из данных, а не их средние значения, что повышает устойчивость к выбросам.
K-Modes — вариант алгоритма K-Means для работы с категориальными данными, который выбирает один из объектов в кластере в качестве моды и минимизирует сумму расстояний Хэмминга между модой и объектами в кластере. Расстояние Хэмминга представляет из себя количество позиций, в которых значения векторов не совпадают.

Одним из ключевых вопросов при использовании K-Means является выбор начальных центроидов, поскольку от них зависит качество и скорость сходимости алгоритма. Существует несколько способов инициализации центроидов:

Случайный выбор: выбирается k случайных точек из данных в качестве начальных центроидов. Такой метод прост, но может привести к плохим результатам, если начальные центроиды слишком близки друг к другу или к краям распределения.
K-Means++: первый центроид выбирается случайно, а затем выбираются остальные центроиды с вероятностью, пропорциональной квадрату расстояния до ближайшего уже выбранного центроида. Данный метод улучшает качество кластеризации, уменьшая вероятность попадания в локальный минимум, но требует дополнительных вычислений.
Greedy K-Means++ — модификация K-Means++, которая ускоряет сходимость и улучшает качество кластеризации за счёт того, что на каждом шаге при выборе центра кластера производится несколько попыток и выбирается лучший (тот, который минимизирует суммарное квадратичное отклонение точек от центров кластеров).

Другим важным аспектом, влияющим на качество алгоритма, является подбор оптимального числа кластеров k, которое устанавливается заранее вручную. Существует несколько методов для выбора оптимального k:

Метод локтя, основанный на графике суммы квадратов расстояний между объектами и центроидами кластеров (SSE) в зависимости от k. Оптимальным k считается та точка, после которой SSE уменьшается незначительно.
Метод силуэта, основанный на графике среднего значения силуэта для каждого k. Силуэт — это мера того, насколько хорошо объект отнесен к своему кластеру по сравнению с другими кластерами. Оптимальным k считается та точка, где силуэт достигает максимума.
Метод комплексной оценки, основанный на анализе нескольких критериев, таких как динамика перераспределения объектов в кластерах, изменение потенциальной энергии объектов внутри кластеров и характерные точки графиков этих критериев. Оптимальным k считается точка, которая удовлетворяет набору правил, использующих данные критерии.

Принцип работы Lloyd's K-Means с инициализацией Greedy K-Means++

Алгоритм строится следующим образом:

1) изначально вычисляются центроиды кластеров с помощью алгоритма Greedy K-Means++;
2) далее рассчитывается квадрат евклидова (или другого) расстояния от каждого наблюдения до центроидов;
3) на основе полученного расстояния наблюдениям присваиваются метки кластеров, которые к ним расположены ближе всего, а также рассчитывается инерция — мера того, насколько хорошо данные были разбиты на кластеры;
4) шаги 2-3 повторяются до тех пор, пока инерция на текущей и предыдущей итерациях перестанет изменяться меньше установленного порога (пока положение центроидов перестанет изменяться в пространстве) или пока не будет достигнуто установленное количество итераций;
5) Наблюдения, расположенные ближе всего к полученным центроидам, и будут составлять кластеры.

Принцип работы алгоритма Greedy K-Means++:

1) первый центроид выбирается случайным образом из данных;
2) для каждого следующего центроида выбирается точка в данных с вероятностью, пропорциональной квадрату расстояния до ближайшего центроида среди нескольких кандидатов, выбранных случайным образом;
3) данный процесс повторяется пока не будет выбрано установленное количество центроидов.

Импорт необходимых библиотек

import numpy as np
from scipy.spatial.distance import cdist
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
from sklearn.metrics import adjusted_rand_score
from sklearn.cluster import KMeans

Реализация на Python с нуля

class KMeansClustering:
    def __init__(self, n_clusters=8, max_iter=300, tol=0.0001, random_state=0):
        self.n_clusters = n_clusters
        self.max_iter = max_iter
        self.tol = tol
        self.random_state = random_state

    def _greedy_kmeans_plus_plus(self, X):
        np.random.seed(self.random_state)
        n_samples, n_features = X.shape
        n_local_trials = 2 + int(np.log(self.n_clusters))

        indices = np.arange(n_samples)
        first_index = np.random.choice(indices)
        centers = np.zeros((self.n_clusters, n_features))

        centers[0] = X[first_index]
        first_center = centers[0].reshape(1, -1)
        sq_distances = cdist(X, first_center, metric='sqeuclidean').ravel()

        for i in range(1, self.n_clusters):
            min_cost = np.inf
            min_new_sq_distances = []
            best_candidate_index = None

            for _ in range(n_local_trials):
                candidates_probas = sq_distances / np.sum(sq_distances)
                candidate_index = np.random.choice(indices, p=candidates_probas)
                candidate = X[candidate_index].reshape(1, -1)

                new_sq_distances = cdist(X, candidate, metric='sqeuclidean').ravel()
                new_cost = np.sum(np.minimum(sq_distances, new_sq_distances))

                if new_cost < min_cost:
                    best_candidate_index = candidate_index
                    min_new_sq_distances = new_sq_distances
                    min_cost = new_cost

            centers[i] = X[best_candidate_index]   # Choose the new center
            sq_distances = np.minimum(sq_distances, min_new_sq_distances)

        return centers

    def fit(self, X):
        n_samples, n_features = X.shape
        self.inertia_ = np.inf
        self.cluster_centers_ = self._greedy_kmeans_plus_plus(X)

        # Lloyd's algorithm
        for _ in range(self.max_iter):
            distances = cdist(X, self.cluster_centers_, metric='sqeuclidean')
            labels = np.argmin(distances, axis=1)
            new_inertia = np.sum(np.min(distances, axis=1))
            new_centers = np.zeros((self.n_clusters, n_features))

            for k in range(self.n_clusters):
                new_centers[k] = np.mean(X[labels == k], axis=0)

            if np.abs(new_inertia - self.inertia_) < self.tol:
                break

            self.inertia_ = new_inertia
            self.cluster_centers_ = new_centers

    def predict(self, X):
        distances = cdist(X, self.cluster_centers_, metric='sqeuclidean')
        predicted_labels = np.argmin(distances, axis=1)

        return predicted_labels

Загрузка датасета

X1, y1 = make_blobs(n_samples=250, n_features=2, centers=8, random_state=0)
print(y1)


[1 3 7 7 6 7 1 3 7 7 0 3 1 1 3 3 5 1 7 4 0 1 1 3 4 7 0 0 6 7 0 0 5 5 7 2 1
 1 6 5 4 7 1 2 1 1 4 3 6 4 7 3 0 2 2 1 7 2 4 0 0 0 1 4 6 5 0 4 6 6 4 4 1 4
 2 3 1 1 5 4 6 4 1 2 5 0 7 6 7 3 0 1 2 5 1 5 3 3 3 1 5 4 0 4 7 6 2 2 2 4 6
 2 5 1 6 4 0 6 5 0 0 6 3 5 1 6 0 2 5 5 6 3 3 1 5 4 5 0 2 2 3 0 4 7 5 4 2 0
 2 6 2 5 2 1 4 1 5 0 4 6 7 5 5 7 6 2 2 3 6 1 7 3 4 7 2 6 6 4 2 2 0 5 4 4 6
 3 1 7 6 7 7 0 4 5 7 2 6 6 2 5 3 3 2 7 1 7 6 6 4 3 5 7 6 3 5 0 3 3 5 5 2 0
 6 3 4 0 5 3 5 2 0 6 4 0 1 1 2 2 0 1 3 7 0 7 0 3 4 7 3 7]

Обучение моделей и оценка полученных результатов

Как можно заметить, K-Means дал достаточно хорошую, но не идеальную кластеризацию данных, что обусловлено наличием небольшого числа выбросов и более сложной формой некоторых кластеров, чем сферическая.

KMeans

kmeans = KMeansClustering(n_clusters=8, random_state=0)
kmeans.fit(X1)
kmeans_pred_res = kmeans.predict(X1)
kmeans_ari = adjusted_rand_score(y1, kmeans_pred_res)
kmeans_centroinds = kmeans.cluster_centers_
print(f'Adjusted Rand Score for KMeans: {kmeans_ari}', '', sep='\n')
print('centroids', kmeans_centroinds, '', sep='\n')
print('prediction', kmeans_pred_res, sep='\n')


Adjusted Rand Score for KMeans: 0.8082423809657193

centroids
[[ 9.20217726 -2.23709633]
 [-1.5438023   7.64224793]
 [-8.61527648 -8.32916569]
 [ 0.81231976  4.02302811]
 [-1.81106448  2.87987747]
 [ 2.3666746   1.30457024]
 [ 1.47433518  8.49698324]
 [ 5.86512606  0.19818122]]

prediction
[5 1 2 2 6 2 5 1 2 2 5 1 5 5 1 1 5 5 2 0 3 7 3 1 0 2 3 3 6 2 1 3 7 7 2 4 5
 5 6 7 0 2 5 4 5 5 0 1 6 7 2 1 3 3 4 5 2 4 0 3 3 3 5 0 6 7 3 0 6 6 0 0 5 0
 3 1 5 5 7 0 3 0 5 4 7 3 2 6 2 6 3 5 4 5 5 7 1 1 1 3 7 0 3 0 2 6 3 4 4 0 6
 4 7 5 1 0 3 6 7 3 3 6 1 7 5 6 3 4 7 7 6 1 1 5 7 0 7 3 4 4 1 3 0 2 7 0 4 3
 4 6 4 7 4 5 0 5 7 3 0 6 2 7 5 2 6 4 3 3 6 5 2 1 0 2 4 6 6 0 4 4 3 7 0 0 6
 1 5 2 6 2 2 3 0 7 2 4 6 6 4 7 1 1 4 2 5 2 6 1 0 1 5 2 6 6 7 3 1 1 7 7 4 3
 6 1 0 3 7 1 7 4 4 6 0 3 5 5 4 4 3 3 1 2 3 2 3 1 0 2 1 2]

KMeans (scikit-learn)

sk_kmeans = KMeans(n_clusters=8, n_init='auto', random_state=0)
sk_kmeans.fit(X1)
sk_kmeans_pred_res = sk_kmeans.predict(X1)
sk_kmeans_ari = adjusted_rand_score(y1, sk_kmeans_pred_res)
sk_kmeans_centroinds = sk_kmeans.cluster_centers_
print(f'Adjusted Rand Score for sk KMeans: {sk_kmeans_ari}', '', sep='\n')
print(sk_kmeans_centroinds, '', sep='\n')
print('prediction', sk_kmeans_pred_res, sep='\n')


Adjusted Rand Score for sk KMeans: 0.8082423809657193

[[ 9.20217726 -2.23709633]
 [-1.5438023   7.64224793]
 [-8.61527648 -8.32916569]
 [ 0.81231976  4.02302811]
 [-1.81106448  2.87987747]
 [ 2.3666746   1.30457024]
 [ 1.47433518  8.49698324]
 [ 5.86512606  0.19818122]]

prediction
[5 1 2 2 6 2 5 1 2 2 5 1 5 5 1 1 5 5 2 0 3 7 3 1 0 2 3 3 6 2 1 3 7 7 2 4 5
 5 6 7 0 2 5 4 5 5 0 1 6 7 2 1 3 3 4 5 2 4 0 3 3 3 5 0 6 7 3 0 6 6 0 0 5 0
 3 1 5 5 7 0 3 0 5 4 7 3 2 6 2 6 3 5 4 5 5 7 1 1 1 3 7 0 3 0 2 6 3 4 4 0 6
 4 7 5 1 0 3 6 7 3 3 6 1 7 5 6 3 4 7 7 6 1 1 5 7 0 7 3 4 4 1 3 0 2 7 0 4 3
 4 6 4 7 4 5 0 5 7 3 0 6 2 7 5 2 6 4 3 3 6 5 2 1 0 2 4 6 6 0 4 4 3 7 0 0 6
 1 5 2 6 2 2 3 0 7 2 4 6 6 4 7 1 1 4 2 5 2 6 1 0 1 5 2 6 6 7 3 1 1 7 7 4 3
 6 1 0 3 7 1 7 4 4 6 0 3 5 5 4 4 3 3 1 2 3 2 3 1 0 2 1 2]

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.subplots_adjust(wspace=0.2)
plt.scatter(X1[:, 0], X1[:, 1], c=kmeans_pred_res, cmap="rainbow")
plt.scatter(kmeans_centroinds[:, 0], kmeans_centroinds[:, 1], marker="x", color="black", s=100)
plt.title("KMeans")
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(1, 2, 2)
plt.subplots_adjust(wspace=0.2)
plt.scatter(X1[:, 0], X1[:, 1], c=sk_kmeans_pred_res, cmap="rainbow")
plt.scatter(sk_kmeans_centroinds[:, 0], sk_kmeans_centroinds[:, 1], marker="x", color="black", s=100)
plt.title("KMeans (scikit-learn)")
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки K-Means

Преимущества:

прост в реализации и понимании;
наличие большого числа модификаций;
высокая скорость работы и точность на данных сферической формы.

Недостатки:

низкая точность на данных с несферической формой кластеров;
чувствительность к начальным значениям центроидов и выбросам;
необходимость заранее устанавливать число кластеров, что может быть сложно или не оптимально.

Дополнительные источники

Статья «Improved Guarantees for k-means++ and k-means++ Parallel», Konstantin Makarychev, Aravind Reddy and Liren Shan.

Документация: описание K-Means, K-Means (алгоритм).

Видео: один, два.

Агломеративная кластеризация

Алгломеративная кластеризация — это метод иерархической кластеризации, который объединяет объекты в кластеры на основе их близости. Процесс кластеризации начинается с того, что изначально каждый объект считается отдельным кластером, а затем на каждом шаге два наиболее близких кластера сливаются в один, пока не будет достигнуто желаемое число кластеров или один общий кластер.

Важным аспектом в алгломеративной кластеризаци является способ вычисления расстояния между кластерами, который называется связью. Можно выделить несколько методов для вычисления связи:

Метод одиночной связи (single linkage): расстояние между кластерами равно минимальному расстоянию между точками из разных кластеров. Этот метод склонен к созданию длинных и извилистых кластеров.

$d_{\min}(C_i, C_j) = \min_{x \in C_i, y \in C_j} \rho(x, y)$

Метод полной связи (complete linkage): расстояние между кластерами равно максимальному расстоянию между точками из разных кластеров. Этот метод склонен к созданию компактных и сферических кластеров.

$d_{\max}(C_i, C_j) = \max_{x \in C_i, y \in C_j} \rho(x, y)$

Метод средней связи (average linkage): расстояние между кластерами равно среднему расстоянию между всеми парами точек из разных кластеров. Этот метод является компромиссом между методами одиночной и полной связи.

$d_{\text{avg}}(C_i, C_j) = \frac{1}{n_i n_j} \sum_{x \in C_i} \sum_{y \in C_j} \rho(x, y)$

Метод Уорда (Ward's linkage): расстояние между кластерами равно приросту суммы квадратов расстояний от точек до центроидов кластеров при объединении этих кластеров. Этот метод стремится минимизировать внутрикластерную дисперсию.

$d_{\text{ward}}(C_i, C_j) = \frac{n_i n_j}{n_i + n_j} \rho^2(\bar{x}_i, \bar{x}_j)$

Принцип работы агломеративной кластеризации на основе метода Уорда

Алгоритм строится следующим образом:

1) для каждой пары кластеров рассчитывается расстояние Уорда на основе евклидова расстояния;
2) метки с минимальным расстоянием, то есть ближайшие кластеры объединяются в новый следующим образом: всем объектам одного кластера присваиваются метки другого, после чего все новые метки, которые больше меток другого кластера, уменьшаются на 1, то есть их нумерация сдвигается влево для устранения пропусков в последовательности;
3) шаги 1-2 повторяются пока количество кластеров больше целевого n_clusters.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial.distance import sqeuclidean
from scipy.cluster.hierarchy import linkage, dendrogram
from sklearn.datasets import make_blobs
from sklearn.metrics import adjusted_rand_score
from sklearn.cluster import AgglomerativeClustering

Реализация на Python с нуля

class HierarchicalAgglomerativeClustering:
    def __init__(self, n_clusters=2):
        self.n_clusters = n_clusters

    @staticmethod
    def _ward_distance(c1, c2):
        n1, n2 = len(c1), len(c2)
        c1_mean, c2_mean = np.mean(c1, axis=0), np.mean(c2, axis=0)
        sqeuclidean_dist = sqeuclidean(c1_mean, c2_mean)

        return (n1 * n2) / (n1 + n2) * sqeuclidean_dist

    @staticmethod
    def _update_labels(labels, min_cdist_idxs):
        # assign a cluster number to labels
        labels[labels == min_cdist_idxs[1]] = min_cdist_idxs[0]
        labels[labels > min_cdist_idxs[1]] -= 1

        return labels

    def fit_predict(self, X):
        labels = np.arange(len(X))
        clusters = [[x] for x in X]

        while len(clusters) > self.n_clusters:
            min_cdist, min_cdist_idxs = np.inf, []

            for i in range(len(clusters) - 1):
                for j in range(i + 1, len(clusters)):
                    cdist = self._ward_distance(clusters[i], clusters[j])

                    if cdist < min_cdist:
                        min_cdist = cdist
                        min_cdist_idxs = (i, j)

            labels = self._update_labels(labels, min_cdist_idxs)
            clusters[min_cdist_idxs[0]].extend(clusters.pop(min_cdist_idxs[1]))

        return np.array(labels)

Загрузка датасета

X2, y2 = make_blobs(n_samples=75, n_features=2, centers=5, random_state=0)
print(y2)


[0 3 4 3 2 4 0 2 0 4 2 4 2 2 0 0 0 3 2 0 2 2 2 3 4 1 1 2 3 0 4 4 3 3 3 2 2
 0 1 1 3 1 0 2 4 1 4 4 0 4 1 0 3 0 4 1 2 1 4 3 1 1 3 0 3 4 1 2 1 4 0 3 1 1
 3]

Обучение моделей и оценка полученных результатов

В данном случае агломеративная кластеризация также справилась достаточно хорошо, но при сравнении графиков можно заметить, что метки ручной реализации имеют другие значения и соответственно цвет на графике. Это связано с другим порядком формирования кластеров: в scikit-learn используется структура данных, известная как linkage tree, позволяющая визуализировать иерархию кластеров с помощью древовидной диаграммы (дендограммы), что может быть полезно при выборе оптимального количества кластеров.

Стоит отметить, что порядок формирования кластеров не влияет на качество кластеризации, поскольку сами метки кластеров разделяются правильно, что видно из одинаковых значений ARI.

AgglomerativeClustering

ac = HierarchicalAgglomerativeClustering(n_clusters=5)
ac_pred_res = ac.fit_predict(X2)
ac_ari = adjusted_rand_score(y2, ac_pred_res)
print(f'Adjusted Rand Score for AgglomerativeClustering: {ac_ari}', '', sep='\n')
print('prediction', ac_pred_res, sep='\n')


Adjusted Rand Score for AgglomerativeClustering: 0.8370870157432925

prediction
[0 1 2 0 3 2 0 3 0 2 3 2 3 3 0 0 0 1 3 0 3 3 0 1 2 4 4 3 1 3 2 2 1 1 1 3 3
 0 4 4 1 4 0 3 2 4 2 2 3 2 4 0 1 0 2 4 3 4 2 1 4 4 1 0 1 2 4 3 3 2 0 1 4 4
 1]

AgglomerativeClustering (scikit-learn)

sk_ac = AgglomerativeClustering(n_clusters=5, linkage='ward')
sk_ac_pred_res = sk_ac.fit_predict(X2)
sk_ac_ari = adjusted_rand_score(y2, sk_ac_pred_res)
print(f'Adjusted Rand Score for sk AgglomerativeClustering: {sk_ac_ari}', '', sep='\n')
print('prediction', sk_ac_pred_res, sep='\n')


Adjusted Rand Score for sk AgglomerativeClustering: 0.8370870157432925

prediction
[4 2 1 4 0 1 4 0 4 1 0 1 0 0 4 4 4 2 0 4 0 0 4 2 1 3 3 0 2 0 1 1 2 2 2 0 0
 4 3 3 2 3 4 0 1 3 1 1 0 1 3 4 2 4 1 3 0 3 1 2 3 3 2 4 2 1 3 0 0 1 4 2 3 3
 2]

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.scatter(X2[:, 0], X2[:, 1], c=ac_pred_res, cmap='rainbow')
plt.title('AgglomerativeClustering')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(1, 2, 2)
plt.scatter(X2[:, 0], X2[:, 1], c=sk_ac_pred_res, cmap='rainbow')
plt.title('AgglomerativeClustering (scikit-learn)')
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

linkage_matrix = linkage(X2, method='ward', metric='euclidean')

plt.figure(figsize=(10, 6))
dendrogram(linkage_matrix, color_threshold=10)
plt.xlabel("Sample Index")
plt.ylabel("Distance")
plt.title("Dendrogram")
plt.show()

Преимущества и недостатки аглометративной кластеризации

Преимущества:

адаптация к различным ситуациям и способность обнаружения кластеров произвольной формы;
возможность формирования информативной иерархии кластеров, которая позволяет выбирать оптимальное число кластеров или лучше изучать структуру данных.

Недостатки:

использование большого количества вычислительных ресурсов и памяти из-за работы со всей матрицей расстояний между объектами;
чувствительность к выбору критерия объединения кластеров, а также неустойчивость к шуму и выбросам, что может сильно искажать иерархию кластеров.

Дополнительные источники

Статья «Modern hierarchical, agglomerative clustering algorithms», Daniel Müllner.

Документация: описание AgglomerativeClustering, AgglomerativeClustering (алгоритм).

Видео: один, два, три.

Спектральная кластеризация

Спектральная кластеризация — метод кластеризации на основе спектральных свойств матрицы сходства графа, который представляет собой набор точек данных, связанных друг с другом в зависимости от их сходства. Основная идея заключается в преобразовании матрицы сходства графа в лаплассиан для получения его собственных векторов, которые в дальнейшем используются для проекции данных в новое пространство более низкой размерности для лучшей разделимости, где затем применяется другой метод кластеризации, например, такой как K-средних.

Существуют различные способы построения матрицы сходства, среди которых наиболее популярными являются следующие:

nearest_neighbors (путем вычисления графа ближайших соседей);
rbf (с использованием радиальной базисной функции);
предварительно вычисленные, где признаки представлены как матрица сходства или граф ближайших соседей;
различные ядра: xi-квадрат, линейное, полиномиальное, сигмоидальное и другие.

Также существуют различные стратегии разложения лаплассиана на собственные вектора и значения:

ARPACK (ARnoldi PACKage) — это программный пакет на языке Fortran, который реализует итерационный метод Арнольди для нахождения нескольких собственных значений и собственных векторов большой разреженной матрицы. Метод Арнольди — это обобщение метода Ланцоша, который строит ортогональный базис подпространства Крылова, порождённого матрицей и начальным вектором, а затем проецирует исходную собственную задачу на это подпространство.
LOBPCG (Locally Optimal Block Preconditioned Conjugate Gradient) — это метод, который реализует локально оптимальный, блочный, предобусловленный, сопряжённый градиент для нахождения нескольких наименьших по модулю собственных значений и собственных векторов большой положительно определенной матрицы. Основная идея LOBPCG заключается в том, что на каждой итерации он выбирает следующее приближение к собственному вектору так, чтобы минимизировать квадратичную форму, связанную с собственной задачей на трехмерном подпространстве, порождённом текущим приближением, предобусловленным остатком и последним обновлением. Это делается с помощью метода Релея-Ритца, который находит оптимальные линейные комбинации этих векторов.
AMG (Algebraic Multigrid methods) — это класс алгоритмов для решения больших систем уравнений, возникающих при дискретизации дифференциальных уравнений. В контексте кластеризации AMG методы уменьшают размер матрицы, переводя её на грубую сетку, где она имеет меньше элементов, но сохраняет свои свойства. Это делается с помощью двух операций: сглаживания и коррекции. Сглаживание — это простой итерационный метод, который уменьшает ошибку в приближённом решении собственной задачи. Коррекция — это переход на грубую сетку, где решается упрощённая собственная задача, а затем возвращается на тонкую сетку для корректировки решения. Этот процесс повторяется несколько раз, пока не будет достигнута нужная точность.

Стоит отметить, что последние 2 метода являются более быстрыми, но менее стабильными.

Принцип работы спектральной кластеризации ARPACK с RBF ядром

Алгоритм строится следующим образом:

1) на основе матрицы сходства графа строится его нормализованный лаплассиан $L = I - D^{-\frac{1}{2}}$ и диагональная матрица степеней вершин графа, где — единичная матрица;
2) далее с помощью алгоритма ARPACK вычисляются k-собственных векторов лаплассиана с указанием сдвига для ускорения вычислений (sigma), случайного вектора (v0), а также наибольших по модулю собственных значений (which="LM") так как они будут расположены ближе всего к сдвигу, что в конечном счёте вернёт наименьшие собственные вектора;
3) полученные вектора нормализуются по степеням вершин графа, чтобы быть независимыми от весов вершин, после чего в даннных векторах изменяется знак, чтобы стать детерменированными (такой подход позволяет избежать неоднозначности в знаках собственных векторов при использовании различных реализаций алгоритма);
4) из модифицированных собственных векторов формируется матрица вложения, к которой применяется алгоритм K-Means, который спрогнозирует в конечном счёте итоговые метки.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from scipy.sparse.linalg import eigsh
from scipy.sparse.csgraph import laplacian as csgraph_laplacian
from sklearn.datasets import make_moons
from sklearn.preprocessing import StandardScaler
from sklearn.cluster import KMeans, SpectralClustering
from sklearn.metrics.pairwise import rbf_kernel
from sklearn.metrics.cluster import adjusted_rand_score

Реализация на Python с нуля

class ArpackSpectralClustering:
    def __init__(self, n_clusters=8, gamma=1.0, random_state=None):
        self.n_clusters = n_clusters
        self.gamma = gamma
        self.random_state = random_state

    @staticmethod
    def _deterministic_vector_sign_flip(u):
        # Flip the sign of the vectors to make them deterministic
        max_abs_rows = np.argmax(np.abs(u), axis=1)
        signs = np.sign(u[range(u.shape[0]), max_abs_rows])

        return u * signs[:, np.newaxis]

    def _spectral_embedding(self, affinity):
        laplacian, diag_vertex_degrees = csgraph_laplacian(affinity, normed=True, return_diag=True)
        laplacian *= -1

        arpack_v0 = np.random.RandomState(self.random_state).uniform(-1, 1, laplacian.shape[0])
        _, evecs = eigsh(laplacian, k=self.n_clusters, sigma=1.0, which="LM", tol=0, v0=arpack_v0)
        norm_evecs = evecs.T[self.n_clusters::-1] / diag_vertex_degrees
        embedding = self._deterministic_vector_sign_flip(norm_evecs)

        return embedding[:self.n_clusters].T

    def fit_predict(self, X):
        affinity_matrix = rbf_kernel(X, gamma=self.gamma)
        embedding = self._spectral_embedding(affinity_matrix)
        kmeans = KMeans(n_clusters=self.n_clusters, n_init='auto', random_state=self.random_state)
        labels = kmeans.fit_predict(embedding)

        return labels

Загрузка датасета

X3, y3 = make_moons(n_samples=200, noise=0.05, random_state=0)
X3 = StandardScaler().fit_transform(X3)
print(y3)


[0 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1
 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1
 1 0 1 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0
 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1
 0 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1
 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1]

Обучение моделей и оценка полученных результатов

Поскольку спектральное преобразование матрицы сходства графа позволяет выявлять скрытую геометрическую структуру данных, алгоритм спектральной кластеризации может применяться к нелинейно разделимым данным и показывать довольно хорошие результаты кластеризации сложных форм, что видно из результатов ниже.

SpectralClustering

sc = ArpackSpectralClustering(n_clusters=2, gamma=10, random_state=0)
sc_pred_res = sc.fit_predict(X3)
sc_ari = adjusted_rand_score(y3, sc_pred_res)
print(f'Adjusted Rand Score for SpectralClustering: {sc_ari}', '', sep='\n')
print('prediction', sc_pred_res, sep='\n')


Adjusted Rand Score for SpectralClustering: 1.0

prediction
[0 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1
 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1
 1 0 1 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0
 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1
 0 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1
 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1]

SpectralClustering (scikit-learn)

sk_sc = SpectralClustering(n_clusters=2, gamma=10, random_state=0)
sk_sc_pred_res = sk_sc.fit_predict(X3)
sk_sc_ari = adjusted_rand_score(y3, sk_sc_pred_res)
print(f'Adjusted Rand Score for sk SpectralClustering: {sk_sc_ari}', '', sep='\n')
print('prediction', sk_sc_pred_res, sep='\n')


Adjusted Rand Score for sk SpectralClustering: 1.0

prediction
[0 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1
 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1
 1 0 1 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0
 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 1
 0 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1
 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1]

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(121)
plt.scatter(X3[:, 0], X3[:, 1], c=sc_pred_res, cmap='Spectral')
plt.title('SpectralClustering')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(122)
plt.scatter(X3[:, 0], X3[:, 1], c=sk_sc_pred_res, cmap='Spectral')
plt.title('SpectralClustering (scikit-learn)')
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки спектральной кластеризации

Преимущества:

работа с кластерами сложных форм;
возможность обработки многомерных данных из-за понижения размерности перед их кластеризацией;
устойчивость к выбросам и шуму в данных из-за учёта их глобальной структуры, а не только локальной.

Недостатки:

сложность конфигурации из-за большого количества гиперпараметров;
высокая вычислительная сложность и потребление памяти при работе с большими объемами данных, что может быть частично решено с помощью безматричных методов или случайных проекций.

Дополнительные источники

Статья «On Spectral Clustering: Analysis and an algorithm», Andrew Y. Ng, Michael I. Jordan, Yair Weiss.

Документация: описание SpectralClustering, SpectralClustering (алгоритм).

Видео: один, два, три.

DBSCAN

Более интересным алгоритмом кластеризации является DBSCAN (Density-Based Spatial Clustering of Applications with Noise), который основан на плотности точек в пространстве. Он группирует вместе точки, которые находятся близко друг к другу и отмечает как выбросы точки, которые лежат в областях с низкой плотностью. Помимо того что DBSCAN может обнаруживать кластеры произвольной формы и выбросы в данных, его главная особенность заключается в самостоятельном определении необходимого количества кластеров, что избавляет от необходимости в их подборе.

Для вычисления попарных расстояний и ближайших соседей точек в DBSCAN используется модифицированная реализация k-ближайших соседей, которая является алгоритмом обучения без учителя и представлена в scikit-learn в виде класса NearestNeighbors.

Схема образования кластера в DBSCAN

Принцип работы DBSCAN

Наиболее важными параметрами, влияющими на результат кластеризации, являются eps (максимально допустимое расстояние между точками, чтобы считаться соседями) и min_samples (минимальное количество точек в окрестности другой точки, чтобы считаться базовой точкой), а сами же точки в данных подразделяются на 3 вида:

Core points (базовые точки) — точки в кластере, имеющие min_samples соседей в своём окружении eps или более. Это означает, что точки лежат в области высокой плотности данных.
Border points (пограничные точки) — точки в кластере, которые имеют меньше, чем min_samples соседей в своём окружении eps, но лежат в окружении eps других базовых точек. Это означает, что точки лежат на границе кластеров.
Noise points (шумовые точки) — это выбросы, которые не принадлежат ни к одному кластеру, то есть точки расположены в области низкой плотности данных.

Алгоритм строится следующим образом:

1) изначально все метки кластеров помечаются как шумовые;
2) после для каждой точки в своём окружении находятся соседи и на их основе отбираются базовые точки;
3) на основе базовых точек и соседей в своём окружении, с помощью метода _dbscan_inner обновляются метки кластеров, которые и будут конечным прогнозом.

_dbscan_inner строится следующим образом:

1) непомеченным базовым точкам присваиваются текущие метки и они добавляются в очередь для дальнейшей обработки;
2) из данной очереди для каждой базовой точки находятся соседи, где каждому непомеченному соседу присваивается текущая метка и он добавляется в очередь из шага 1;
3) далее текущая метка увеличивается на 1 и шаг 2 повторяется пока очередь не станет пустой.

В результате данного процесса каждой точке присваивается метка соответствующего кластера, к которому она принадлежит. Точки, которые не принадлежат ни к одному кластеру, остаются шумовыми с меткой -1.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import DBSCAN
from sklearn.neighbors import NearestNeighbors
from sklearn.datasets import make_circles
from sklearn.metrics import adjusted_rand_score
from sklearn.preprocessing import StandardScaler

Реализация на Python с нуля

class DBSCANClustering:
    def __init__(self, eps=0.5, min_samples=5, metric='euclidean', algorithm='auto', leaf_size=30):
        self.eps = eps
        self.min_samples = min_samples
        self.metric = metric
        self.algorithm = algorithm
        self.leaf_size = leaf_size

    @staticmethod
    def _dbscan_inner(core_points, neighborhoods, labels):
        label, queue = 0, []

        for point in range(len(core_points)):
            # if the point is already assigned a label or not a core point, skip it
            if not core_points[point] or labels[point] != -1:
                continue

            labels[point] = label
            queue.append(point)

            while queue:
                current_point = queue.pop(0)
                # if the point is a core point, get it's neighbors
                if core_points[current_point]:
                    current_neighbors = neighborhoods[current_point]

                    for neighbor in current_neighbors:
                        if labels[neighbor] == -1:
                            labels[neighbor] = label
                            queue.append(neighbor)

            label += 1

    def fit_predict(self, X):
        nn = NearestNeighbors(n_neighbors=self.min_samples, radius=self.eps, metric=self.metric,
                              algorithm=self.algorithm, leaf_size=self.leaf_size)

        # find the neighbors for each point within the given radius
        neighborhoods = nn.fit(X).radius_neighbors(X, return_distance=False)
        labels = np.full(len(X), -1, dtype=np.intp)
        core_points = np.asarray([len(n) >= self.min_samples for n in neighborhoods])

        self._dbscan_inner(core_points, neighborhoods, labels)
        self.labels_ = labels

        return self.labels_

Загрузка датасета

X4, y4 = make_circles(n_samples=250, noise=0.05, factor=0.5, random_state=0)
X4 = StandardScaler().fit_transform(X4)
print(y4)


[1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1
 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 0
 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 0
 1 1 1 0 0 0 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 0 1 1 0 0 0 0 0
 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1
 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1
 0 0 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1]

Обучение моделей и оценка полученных результатов

Как можно заметить, DBSCAN также отлично справился с кластеризацией данных сложной формы, однако выбор оптимальных eps и min_samples на практике может оказаться довольно трудной задачей, поскольку данные параметры существенно влияют на результаты кластеризации.

Частично данную проблему можно решить с использованием HDBSCAN — модификации DBSCAN, которая автоматически находит подходящее значение eps для каждого кластера, используя иерархический подход, что позволяет определять кластеры разной плотности и повысить устойчивость к выбросам.

Ещё одной интересной модификацией, похожей на HDBSCAN, является OPTICS (Ordering Points To Identify the Clustering Structure), где используется граф достижимости, который определяет достижимое расстояние для каждой точки, которая в дальнейшем будет относиться к ближайшему кластеру. Такой подход позволяет ещё лучше определять кластеры разной плотности, особенно если они расположены близко друг к другу, однако это увеличивает время работы алгоритма.

DBSCAN

dbscan = DBSCANClustering(eps=0.3, min_samples=3)
dbscan_pred_res = dbscan.fit_predict(X4)
dbscan_ari = adjusted_rand_score(y4, dbscan_pred_res)
print(f'Adjusted Rand Score for DBSCAN: {dbscan_ari}', '', sep='\n')
print('prediction', dbscan_pred_res, sep='\n')


Adjusted Rand Score for DBSCAN: 1.0

prediction
[0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0
 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1
 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1
 0 0 0 1 1 1 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1
 1 1 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 0 1 1 0
 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0
 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 1 0]

DBSCAN (scikit-learn)

sk_dbscan = DBSCAN(eps=0.3, min_samples=3)
sk_dbscan_pred_res = sk_dbscan.fit_predict(X4)
sk_dbscan_ari = adjusted_rand_score(y4, sk_dbscan_pred_res)
print(f'Adjusted Rand Score for sk DBSCAN: {sk_dbscan_ari}', '', sep='\n')
print('prediction', sk_dbscan_pred_res, sep='\n')


Adjusted Rand Score for sk DBSCAN: 1.0

prediction
[0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0
 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 1
 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1
 0 0 0 1 1 1 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1
 1 1 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 0 1 1 0
 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0
 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 1 0]

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.scatter(X4[:, 0], X4[:, 1], c=dbscan_pred_res, cmap='rainbow')
plt.title('DBSCAN')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(1, 2, 2)
plt.scatter(X4[:, 0], X4[:, 1], c=sk_dbscan_pred_res, cmap='rainbow')
plt.title('DBSCAN (scikit-learn)')
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки DBSCAN

Преимущества:

устойчив к выбросам;
не требуется заранее указывать количество кластеров;
способность находить кластеры произвольной формы, а также шумные точки в данных.

Недостатки:

плохая работа с кластерами разной плотности;
требуется большой объём памяти для хранения расстояний между всеми точками;
высокая чувствительность к выбору параметров eps и min_samples, что может сильно повлиять на качество кластеризации в негативную сторону.

Дополнительные источники

Статьи:

«DBSCAN: Optimal Rates For Density-Based Cluster Estimation», Daren Wang, Xinyang Lu and Alessandro Rinaldo;
«HDBSCAN: Density based Clustering over Location Based Services», Md Farhadur Rahman, Weimo Liu Saad Bin Suhaim, Saravanan Thirumuruganathan, Nan Zhang, Gautam Das;
«OPTICS: Ordering Points To Identify the Clustering Structure», Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel, Jörg Sander.

Документация:

описание DBSCAN ;
описание HDBSCAN ;
описание OPTICS ;
DBSCAN (алгоритм);
HDBSCAN (алгоритм);
OPTICS (алгоритм).

Видео:

DBSCAN: один, два, три.
HDBSCAN: один, два.
OPTICS: один, два.

Affinity Propagation

Ещё более продвинутым подходом относительно предыдущего алгоритма кластеризации является метод распространения близости (affinity propagation), который основан на концепции соотношения между данными и выборе из них экземпляров — наиболее репрезентативных образцов, которые представляются центроидами кластеров и группируют возле себя все остальные данные.

Соотношения между данными описываются с помощью матриц сходства S (similarity matrix), доступности A (availability matrix) и ответственности R (responsibility matrix), а наиболее важными параметрами при настройке алгоритма являются damping (фактор затухания, который не дает алгоритму слишком быстро менять своё мнение о том, какие точки данных лучше всего подходят друг другу) и preference (мера предпочтения точки быть экземпляром для себя или для других точек: чем больше это значение, тем больше вероятность быть экземпляром).

Принцип работы Affinity Propagation

Рассмотрим следующий пример для лучшего понимания сути данного алгоритма на интуитивном уровне. Представьте, что в школе есть много учеников, которые хотят найти своих друзей. Каждый ученик имеет свои предпочтения с кем он хочет дружить, исходя из общих интересов, характера, внешности и так далее. Например, один ученик любит футбол, другой ученик любит музыку, третий ученик любит математику. Эти предпочтения можно измерить числом, которое показывает насколько сильно ученик хочет дружить с другим учеником. Это называется сходством: чем оно больше, тем больше шансов, что ученики станут друзьями.

В данном случае Affinity propagation — это алгоритм, который помогает ученикам найти своих друзей на основе их предпочтений следующим образом:

Сначала каждый ученик оценивает насколько он хочет дружить с другими учениками. Например, ученик A может сказать, что он хочет дружить с учеником B на 8 из 10, с учеником C на 6 из 10, с учеником D на 4 из 10 и так далее. Эти оценки можно рассчитать с помощью отрицательного квадратичного евклидова расстояния и записать в виде матрицы сходства, где каждая строка и столбец соответствует ученику, а каждая ячейка содержит сходство между двумя учениками.

Затем каждый ученик отправляет сообщения другим ученикам, в которых говорит насколько он хочет, чтобы они были его друзьями. Это называется ответственностью. Ответственность — это степень с которой ученик выбирает другого ученика в качестве своего друга. Ответственность зависит не только от сходства, но и от того насколько другие ученики хотят быть друзьями с тем же учеником. Например, если ученик A хочет дружить с учеником B, но ученик B не хочет дружить с учеником A, то ответственность ученика A к ученику B будет низкой. Ну а если ученик A хочет дружить с учеником C, и ученик C тоже хочет дружить с учеником A, то ответственность ученика A к ученику C будет высокой. Ответственность можно вычислить по формуле:

$r(i, k) = s(i, k) - \max_{k' \neq k} \{ a(i, k') + s(i, k') \}$

где r(i, k) — это ответственность ученика i к ученику k, s(i, k) — это сходство ученика i к ученику k, a(i, k') — это доступность ученика i к ученику k', которая будет объяснена далее. Ответственность можно также записать в виде матрицы, где каждая строка и столбец соответствует ученику, а каждая ячейка содержит ответственность между двумя учениками.

Далее каждый ученик получает сообщения от других учеников, в которых они говорят насколько они хотят, чтобы он был их другом. Это называется доступностью. Доступность — это степень с которой ученик подходит для роли друга для другого ученика. Доступность зависит не только от ответственности, но и от того, насколько другие ученики подходят для роли друзей для того же ученика. Например, если ученик A хочет дружить с учеником B и ученик B хочет дружить с учеником A, то доступность ученика A к ученику B будет высокой. Ну а если ученик A хочет дружить с учеником C, но ученик C не хочет дружить с учеником A, то доступность ученика A к ученику C будет низкой. Доступность можно вычислить по формуле:

$a(i, k) = \min \{ 0, r(k, k) + \sum_{i' \neq i, k} \max \{ 0, r(i', k) \} \}$

где a(i, k) — это доступность ученика i к ученику k, r(k, k) — это ответственность ученика k к себе, r(i', k) - это ответственность другого ученика i' к ученику k. Доступность можно также записать в виде матрицы, где каждая строка и столбец соответствует ученику, а каждая ячейка содержит доступность между двумя учениками.

Алгоритм повторяет эти шаги, пока не найдет оптимальное решение, в котором каждый ученик имеет своего друга. Этот друг называется экземпляром. Экземпляры — это ученики, которые лучше всего подходят для роли друзей для других учеников. Количество экземпляров зависит от того, какие предпочтения задаются для учеников. Если предпочтения не задаются, то они равны медиане сходств. Экземпляры можно определить по формуле:

$e(i) = \arg \max_k \{ r(i, k) + a(i, k) \}$

где e(i) — это экземпляр ученика i, r(i, k) — это ответственность ученика i к ученику k, a(i, k) — это доступность ученика i к ученику k. Эта формула означает, что ученик i выбирает в качестве своего друга того ученика k, для которого сумма ответственности и доступности максимальна. Если ученик i выбирает себя в качестве своего друга, то он является экземпляром. Если ученик i выбирает другого ученика k в качестве своего друга, то он принадлежит к тому же кластеру, что и ученик k. Кластер — это группа учеников, которые имеют одного и того же друга-экземпляра.

Данный алгоритм строится следующим образом:

1) на основе отрицательного квадратичного расстояния находится матрица сходства, а также нулями инициализируются матрицы доступности и ответственности;
2) из матрицы сходства удаляются вырождения, а также добавляется небольшой шум;
3) далее итеративно обновляются значения матриц доступности и ответственности на основе временной матрицы, представленной изначально как сумма матриц сходства и доступности, чтобы найти максимальное сходство между точками данных и потенциальными экземплярами;
4) из этой матрицы вычисляются максимальные значения по столбцам, а также вторые по величине значения, которые используются для вычитания из матрицы сходства, чтобы получить новую матрицу ответственности;
5) при обновлении значений матрицы доступности, временная матрица заполняется положительными значениями матрицы ответственности, чтобы посчитать сумму сообщений между точками о том, насколько сильно они хотят быть экземплярами.
6) затем из данной матрицы вычитается её сумма по столбцам, чтобы получить отрицательную матрицу доступности, а также временная матрица обрезается по нулю для получения положительной матрицы доступности;
7) также при обновлении значений матрицы доступности и ответственности к временной матрице применяется коэффициент затухания для избегания численных колебаний при обновлении значений;
8) экземпляры с положительной суммой ответственности и доступности становятся центрами кластеров;
9) после каждой итерации проверяется условие сходимости алгоритма с использованием матрицы сходимости экземпляров: если число экземпляров не меняется в течение заданного числа итераций или если число итераций достигает максимума, то алгоритм останавливается;
10) в конце уточняется итоговый набор экземпляров и меток кластеров через выбор лучших экземпляров для каждого кластера из его членов на основе максимальной суммы сходств между ними;
11) полученные метки кластеров и будут итоговым прогнозом.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
from sklearn.cluster import AffinityPropagation
from sklearn.metrics import euclidean_distances, adjusted_rand_score

Реализация на Python с нуля

class AffinityPropagationClustering:
    def __init__(self, damping=0.5, max_iter=200, convergence_iter=15, preference=None, random_state=0):
        self.damping = damping
        self.max_iter = max_iter
        self.convergence_iter = convergence_iter
        self.preference = preference
        self.random_state = random_state

    @staticmethod
    def _affinity_propagation_inner(similarity_matrix, preference, convergence_iter, max_iter,
                                   damping, random_state):
        rng = np.random.RandomState(random_state)
        n_samples = similarity_matrix.shape[0]
        samples_indexes = np.arange(n_samples)

        # place preference on the diagonal of similarity matrix
        similarity_matrix.flat[:: (n_samples + 1)] = preference
        availability_matrix = np.zeros((n_samples, n_samples))
        responsibility_matrix = np.zeros((n_samples, n_samples))  # initialize messages
        exemplars_convergence_matrix = np.zeros((n_samples, convergence_iter))

        # remove degeneracies
        similarity_matrix += (np.finfo(similarity_matrix.dtype).eps * similarity_matrix +
                              np.finfo(similarity_matrix.dtype).tiny * 100) * \
                              rng.standard_normal(size=(n_samples, n_samples))

        for iter in range(max_iter):
            temp_matrix = availability_matrix + similarity_matrix   # compute responsibilities
            max_indexes = np.argmax(temp_matrix, axis=1)
            max_values = temp_matrix[samples_indexes, max_indexes]
            temp_matrix[samples_indexes, max_indexes] = -np.inf
            second_max_values = np.max(temp_matrix, axis=1)

            # temp_matrix = new_responsibility_matrix
            np.subtract(similarity_matrix, max_values[:, None], temp_matrix)
            max_responsibility = similarity_matrix[samples_indexes, max_indexes] - second_max_values
            temp_matrix[samples_indexes, max_indexes] = max_responsibility

            # damping
            temp_matrix *= 1 - damping
            responsibility_matrix *= damping
            responsibility_matrix += temp_matrix

            # temp_matrix = Rp; compute availabilities
            np.maximum(responsibility_matrix, 0, temp_matrix)
            temp_matrix.flat[:: n_samples + 1] = responsibility_matrix.flat[:: n_samples + 1]

            # temp_matrix = -new_availability_matrix
            temp_matrix -= np.sum(temp_matrix, axis=0)
            diag_availability_matrix = np.diag(temp_matrix).copy()
            temp_matrix.clip(0, np.inf, temp_matrix)
            temp_matrix.flat[:: n_samples + 1] = diag_availability_matrix

            # damping
            temp_matrix *= 1 - damping
            availability_matrix *= damping
            availability_matrix -= temp_matrix

            # check for convergence
            exemplar = (np.diag(availability_matrix) + np.diag(responsibility_matrix)) > 0
            exemplars_convergence_matrix[:, iter % convergence_iter] = exemplar
            n_exemplars = np.sum(exemplar, axis=0)

            if iter >= convergence_iter:
                exemplars_sum = np.sum(exemplars_convergence_matrix, axis=1)
                unconverged = np.sum((exemplars_sum == convergence_iter) +
                                     (exemplars_sum == 0)) != n_samples

                if (not unconverged and (n_exemplars > 0)) or (iter == max_iter):
                    break

        exemplar_indixes = np.flatnonzero(exemplar)
        n_exemplars = exemplar_indixes.size  # number of detected clusters

        if n_exemplars > 0:
            cluster_indices = np.argmax(similarity_matrix[:, exemplar_indixes], axis=1)
            cluster_indices[exemplar_indixes] = np.arange(n_exemplars)  # Identify clusters

            # refine the final set of exemplars and clusters and return results
            for k in range(n_exemplars):
                cluster_members = np.where(cluster_indices == k)[0]
                best_k = np.argmax(np.sum(similarity_matrix[cluster_members[:, np.newaxis],
                                                            cluster_members], axis=0))
                exemplar_indixes[k] = cluster_members[best_k]

            cluster_indices = np.argmax(similarity_matrix[:, exemplar_indixes], axis=1)
            cluster_indices[exemplar_indixes] = np.arange(n_exemplars)
            labels = exemplar_indixes[cluster_indices]

            # Reduce labels to a sorted, gapless, list
            cluster_centers_indices = np.unique(labels)
            labels = np.searchsorted(cluster_centers_indices, labels)
        else:
            cluster_centers_indices = []
            labels = np.array([-1] * n_samples)

        return cluster_centers_indices, labels

    def fit_predict(self, X):
        self.affinity_matrix_ = -euclidean_distances(X, squared=True)

        if self.preference is None:
            self.preference = np.median(self.affinity_matrix_)

        params = (self.affinity_matrix_, self.preference, self.convergence_iter, self.max_iter,
                  self.damping, self.random_state)

        self.cluster_centers_indices_, self.labels_ = self._affinity_propagation_inner(*params)
        self.cluster_centers_ = X[self.cluster_centers_indices_]

        return self.labels_

Код для отрисовки графика

def plot_connected_points(X, labels, centers, cmap):
    for i in range(len(X)):
        color = cmap(labels[i] / len(centers))
        plt.plot([X[i, 0], centers[labels[i], 0]], [X[i, 1], centers[labels[i], 1]], c=color, alpha=0.8)

Загрузка датасета

X5, y5 = make_blobs(n_samples=300, centers=9, cluster_std=0.5, random_state=0)
print(y5)


[1 1 1 6 5 7 0 3 2 5 8 4 1 2 0 1 6 2 1 6 6 0 6 2 5 5 5 8 8 4 7 3 8 8 0 7 4
 6 2 0 4 4 7 1 7 5 8 0 5 4 0 3 8 8 2 2 4 7 6 7 4 5 4 5 3 8 4 8 8 4 6 6 2 3
 7 7 4 8 4 1 4 8 7 3 7 8 6 1 3 7 5 0 8 3 5 2 4 2 2 2 7 4 6 2 2 7 5 5 7 2 7
 0 5 7 1 5 3 2 8 3 1 4 2 1 8 5 1 5 5 2 2 1 6 7 4 3 3 3 5 3 6 6 6 5 4 4 2 8
 6 5 7 0 8 8 8 1 6 1 6 6 0 7 7 6 0 8 0 3 6 4 7 3 1 6 0 0 4 1 5 6 1 2 1 0 8
 3 0 7 5 4 3 0 5 2 1 4 6 1 0 0 0 3 8 5 0 6 0 1 5 4 8 2 4 0 1 6 3 7 0 8 4 8
 1 8 1 5 0 0 7 5 4 4 3 0 7 3 2 4 3 5 0 7 6 1 0 7 1 3 5 1 7 3 6 7 6 2 1 3 8
 5 2 3 1 4 6 2 3 8 0 1 6 2 7 6 0 7 1 4 3 5 4 0 2 2 2 1 8 8 7 3 2 0 3 2 3 6
 5 8 2 3]

Обучение моделей и оценка полученных результатов

Affinity Propagation показал отличный результат без настройки параметров, определив верно не только количество кластеров, но и присвоив метки всем точкам данных. Данный алгоритм также способен показывать хорошие результаты для кластеров любой формы и плотности, однако может потребоваться настройка параметров, но это всё равно проще, чем в случае с DBSCAN или HDBSCAN.

AffinityPropagation

ap = AffinityPropagationClustering()
ap_pred_res = ap.fit_predict(X5)
ap_ari = adjusted_rand_score(y5, ap_pred_res)
print(f'Adjusted Rand Score for AffinityPropagation: {ap_ari}', '', sep='\n')
print('prediction', ap_pred_res, sep='\n')


Adjusted Rand Score for AffinityPropagation: 1.0

prediction
[1 1 1 7 6 0 8 5 2 6 3 4 1 2 8 1 7 2 1 7 7 8 7 2 6 6 6 3 3 4 0 5 3 3 8 0 4
 7 2 8 4 4 0 1 0 6 3 8 6 4 8 5 3 3 2 2 4 0 7 0 4 6 4 6 5 3 4 3 3 4 7 7 2 5
 0 0 4 3 4 1 4 3 0 5 0 3 7 1 5 0 6 8 3 5 6 2 4 2 2 2 0 4 7 2 2 0 6 6 0 2 0
 8 6 0 1 6 5 2 3 5 1 4 2 1 3 6 1 6 6 2 2 1 7 0 4 5 5 5 6 5 7 7 7 6 4 4 2 3
 7 6 0 8 3 3 3 1 7 1 7 7 8 0 0 7 8 3 8 5 7 4 0 5 1 7 8 8 4 1 6 7 1 2 1 8 3
 5 8 0 6 4 5 8 6 2 1 4 7 1 8 8 8 5 3 6 8 7 8 1 6 4 3 2 4 8 1 7 5 0 8 3 4 3
 1 3 1 6 8 8 0 6 4 4 5 8 0 5 2 4 5 6 8 0 7 1 8 0 1 5 6 1 0 5 7 0 7 2 1 5 3
 6 2 5 1 4 7 2 5 3 8 1 7 2 0 7 8 0 1 4 5 6 4 8 2 2 2 1 3 3 0 5 2 8 5 2 5 7
 6 3 2 5]

AffinityPropagation (scikit-learn)

sk_ap = AffinityPropagation()
sk_ap_pred_res = sk_ap.fit_predict(X5)
sk_ap_ari = adjusted_rand_score(y5, sk_ap_pred_res)
print(f'Adjusted Rand Score for sk AffinityPropagation: {sk_ap_ari}', '', sep='\n')
print('prediction', sk_ap_pred_res, sep='\n')


Adjusted Rand Score for sk AffinityPropagation: 1.0

prediction
[1 1 1 7 6 0 8 5 2 6 3 4 1 2 8 1 7 2 1 7 7 8 7 2 6 6 6 3 3 4 0 5 3 3 8 0 4
 7 2 8 4 4 0 1 0 6 3 8 6 4 8 5 3 3 2 2 4 0 7 0 4 6 4 6 5 3 4 3 3 4 7 7 2 5
 0 0 4 3 4 1 4 3 0 5 0 3 7 1 5 0 6 8 3 5 6 2 4 2 2 2 0 4 7 2 2 0 6 6 0 2 0
 8 6 0 1 6 5 2 3 5 1 4 2 1 3 6 1 6 6 2 2 1 7 0 4 5 5 5 6 5 7 7 7 6 4 4 2 3
 7 6 0 8 3 3 3 1 7 1 7 7 8 0 0 7 8 3 8 5 7 4 0 5 1 7 8 8 4 1 6 7 1 2 1 8 3
 5 8 0 6 4 5 8 6 2 1 4 7 1 8 8 8 5 3 6 8 7 8 1 6 4 3 2 4 8 1 7 5 0 8 3 4 3
 1 3 1 6 8 8 0 6 4 4 5 8 0 5 2 4 5 6 8 0 7 1 8 0 1 5 6 1 0 5 7 0 7 2 1 5 3
 6 2 5 1 4 7 2 5 3 8 1 7 2 0 7 8 0 1 4 5 6 4 8 2 2 2 1 3 3 0 5 2 8 5 2 5 7
 6 3 2 5]

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(121)
plt.scatter(X5[:, 0], X5[:, 1], c=ap_pred_res, cmap='rainbow', s=10)
plt.scatter(ap.cluster_centers_[:, 0], ap.cluster_centers_[:, 1], c='black', s=50)
plt.title('AffinityPropagation')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plot_connected_points(X5, ap_pred_res, ap.cluster_centers_, plt.cm.rainbow)

plt.subplot(122)
plt.scatter(X5[:, 0], X5[:, 1], c=sk_ap_pred_res, cmap='rainbow', s=10)
plt.scatter(sk_ap.cluster_centers_[:, 0], sk_ap.cluster_centers_[:, 1], c='black', s=50)
plt.title('AffinityPropagation (scikit-learn)')
plt.xlabel("Feature 1")
plot_connected_points(X5, sk_ap_pred_res, sk_ap.cluster_centers_, plt.cm.rainbow)

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки Affinity Propagation

Преимущества:

высокая точность;
автоматическое определение количества кластеров;
не чувствителен к выбору начальных значений;
небольшое количество гиперпараметров для настройки.

Недостатки:

возможна чувствительность к выбросам и шуму в данных;
высокая сложность по времени выполнения и использованию памяти, что делает затруднительным его применение к данным большого размера.

Не смотря на то, что Affinity Propagation является далеко не самым быстрым алгоритмом, на данный момент существует несколько модификаций, способных значительно повысить его скорость. Среди наиболее интересных модификаций можно выделить следующие:

Hierarchical Affinity Propagation (HAP) — алгоритм кластеризации с использованием Affinity Propagation несколько раз на разных уровнях данных. Сначала находятся экземпляры на небольшом подмножестве данных, которые после применяются ко всему набору данных. Такой подход позволяет сократить время выполнения и уменьшить количество сообщений, передаваемых между точками, однако это может повлиять на качество кластеризации.
Fast Affinity Propagation — модификация на основе обрезаний сообщений, которая базируется на двух основных идеях. Во-первых, сокращается размер матрицы сходства путем выбора небольшого подмножества экземпляров в качестве кандидатов в прототипы, а также вычисления сходства только между ними и всеми остальными экземплярами. Во-вторых, используется многомерный поиск для определения параметра предпочтения путем разбиения интервала возможных значений предпочтения на несколько подынтервалов и запуска алгоритма на каждом из них параллельно. Затем выбирается тот подынтервал, который дает наилучшее качество кластеризации по некоторому установленному критерию.
Affinity Propagation на основе пиков плотности — двухэтапный алгоритм кластеризации (DDAP), который сначала определяет пики плотности в данных с помощью алгоритма DDC (Density peaks and Distance-based Clustering), а затем использует стандартный Affinity Propagation для поиска экземпляров, которые близки к этим пикам. Такой подход позволяет значительно уменьшить количество вычислений при сопоставимом качестве кластеризации.

Дополнительные источники

Статьи:

«Extended Affinity Propagation: Global Discovery and Local Insights», Rayyan Ahmad Khan, Rana Ali Amjad, Martin Kleinsteuber;
«Hierarchical Affinity Propagation», Inmar E. Givoni, Clement Chung, Brendan J. Frey;
«Fast Affinity Propagation Clustering based on Machine Learning», Shailendra Kumar Shrivastava, Dr. J.L. Rana and Dr. R.C. Jain;
«Fast Clustering by Affinity Propagation Based on Density Peaks», Yang Li, Chonghui Guo, Leilei Sun.

Документация:

описание Affinity Propagation, Affinity Propagation (алгоритм).

Видео: один, два.

The end

Вот и подошёл к концу обзор данной темы. Очень надеюсь, что вам понравилось, поскольку в дальнейшем я планирую опубликовать в подобном формате статьи по всем популярным алгоритмам машинного обучения с их реализацией с нуля на Python.

Всем успехов!

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8 мар. 2024 г.

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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
09.08.25 14:24 lisa111

Recovering lost money from scams calls for cooperation. Recovery professionals locate stolen funds by applying their expertise. Experts in law offer advice on the best course of action. This helps you recoup your investment. Scams teach us to be aware of the warning indications. Promises of large, rapid riches with minimal risk should be avoided. They may want you to make an investment right away. Get advice from Marie on how to prevent frauds in the future. She can help you recover your missing bitcoin as well. You can reach her at [email protected] or @Marie_consultancy on Telegram. Additionally, you can utilize WhatsApp at +1 7127594675.
09.08.25 16:31 vallatjosette

My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
09.08.25 16:43 meghanmarkle1998

I was scammed out $ 180,000 in BTC, but Spikes Assets Recovery came to my rescue. When my cryptocurrency investment fell into the wrong hands, I lost $115,330 in Bitcoin and was defrauded of my hard-earned money. I was on the verge of giving up after learning that the funds seemed unrecoverable, but then I came across a Google post about a legitimate recovery service called Spikes Assets Recovery, a firm specializing in stolen fund and crypto recovery. To my astonishment, after working with them, they successfully recovered the full $115,330 I had lost in BTC. I'm incredibly grateful and relieved that I was able to recover all of my missing Bitcoin. For inquiries or assistance with recovering your lost assets, you can contact Spikes Assets Recovery using the following details: Email: [email protected] Yahoo: [email protected]
09.08.25 17:24 Sophie Pugh

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10.08.25 03:53 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram user name (@RoseCyberHives) and email rosecryptoinvestment11@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram user name (@RoseCyberHives) rosecryptoinvestment11@ gmail. com
10.08.25 03:53 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram user name (@RoseCyberHives) and email rosecryptoinvestment11@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram user name (@RoseCyberHives) rosecryptoinvestment11@ gmail. com
10.08.25 03:54 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram user name (@RoseCyberHives) and email rosecryptoinvestment11@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram user name (@RoseCyberHives) rosecryptoinvestment11@ gmail. com
10.08.25 03:54 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram user name (@RoseCyberHives) and email rosecryptoinvestment11@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram user name (@RoseCyberHives) rosecryptoinvestment11@ gmail. com
10.08.25 03:54 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram user name (@RoseCyberHives) and email rosecryptoinvestment11@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram user name (@RoseCyberHives) rosecryptoinvestment11@ gmail. com
10.08.25 03:54 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram user name (@RoseCyberHives) and email rosecryptoinvestment11@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram user name (@RoseCyberHives) rosecryptoinvestment11@ gmail. com
10.08.25 03:55 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram user name (@RoseCyberHives) and email rosecryptoinvestment11@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram user name (@RoseCyberHives) rosecryptoinvestment11@ gmail. com
10.08.25 04:55 lisared

I thought losing £85,000 in Doge was the end until Agent Jasmine Lopez proved me wrong. With unmatched skill and persistence, she traced my stolen money, tracked down the scammers, and recovered a huge portion of my funds. If you’ve been scammed, don’t wait reach out to her at (recoveryfundprovider@gmail. com) WhatsApp at +{44 736-644-5035}. She could be the reason you get your money back.
10.08.25 12:57 Marthaamahle

My name is Martha, and I never thought I’d be the victim of a crypto scam, but life has a way of humbling you. A few months ago, I invested a huge chunk of my savings, decades of hard-earned money into what seemed like a promising crypto platform. The slick website, glowing testimonials, and a charming advisor convinced me to transfer 15 Bitcoin, into their wallet. Days later, the platform vanished, my funds were gone, and I was left with nothing but regret. Desperate, I turned to the internet, scouring forums and posts on X for help. That’s when I stumbled across Alpha Spy Nest, a shadowy group of cyber-recovery experts specializing in crypto fraud. Their reputation was murky but promising whispers of successful recoveries and a no-nonsense approach. I reached out through their encrypted portal, half-expecting a scam. Within hours, they responded, asking for details: transaction IDs, wallet addresses, and screenshots of my interactions with the scammers. Alpha Spy Nest operated in the dark corners of the web, using a mix of blockchain forensics, social engineering, and unconventional methods to track lost crypto. They traced my Bitcoin to a series of obfuscated wallets, likely controlled by an offshore syndicate. The team deployed advanced chain-analysis tools to follow the funds, pinpointing a weak link in the scammers’ network, a poorly secured exchange account in Eastern Europe.Over weeks, Alpha Spy Nest worked tirelessly, infiltrating the scammers’ digital trails. And then i received a message, we’ve got a lead. They’d recovered 12 of my 15 Bitcoin, transferred to a secure wallet they set up for me. The rest, they said, was likely gone forever, but I was overjoyed. My savings were partially restored, and I felt a weight lift. I never met them, but they gave me back hope and a lesson, the crypto world is a jungle, but sometimes, even in the dark, there are allies. Contact them via: website: www.alphaspynest.org, whatsapp: ‪+15132924878‬ , telegram: https://t.me/Alphaspynest
10.08.25 14:33 tomcooper0147

Losing €256,000 On trading was one of the worst moments of my life. I felt sick, helpless, and convinced I’d never see that money again. Every day that passed made the loss feel heavier. Then I found Agent Jasmine Lopez and everything changed. With relentless determination, she traced the stolen funds, tracked down the scammers, and recovered a huge portion of my losses in the space of a few days. Her skill and persistence gave me hope when I had none. Time is everything contact her now: [email protected] Whats//App at +{44 736-644-5035}.
10.08.25 17:06 kevin

I see a lot of recommendations online and it’s already obvious there are bad eggs online who will only add to your mystery. I can only recommend one and you can reach them via mail on (zattechrecovery AT gmailc0 m ) if you need help on recovering what you lost to scammers.
10.08.25 18:13 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
10.08.25 18:13 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
10.08.25 20:35 Bartlevi

Sylvester demonstrated truly exceptional skill. He successfully recovered over $610,000 in USDT. This was a momentous personal triumph. His support extended far beyond this initial success. He diligently assisted in reclaiming additional lost funds. Throughout this complex process, he maintained consistent contact. He kept me fully updated at every turn. The entire undertaking was conducted with utmost honesty. There were no concealed charges or dubious methods. My funds were directly returned to my crypto wallet. The transaction proceeded smoothly and without any issues. I can confidently state Sylvester Bryant is the most reliable recovery agent I have ever encountered. I strongly urge anyone facing a similar situation to contact him without delay. You may reach Sylvester by email at yt7cracker@gmail. com. You can also contact him on WhatsApp at +1 (512) 577-7957. Do not surrender to despair. Sylvester possesses the ability and commitment to retrieve your stolen funds. He can achieve this for you, as he did for me. Your financial recovery is attainable. Sylvester Bryant is your path to recovering your money.
10.08.25 23:28 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
10.08.25 23:29 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.08.25 10:33 Morris Jason

HIRE A REPUTABLE RECOVERY FIRM THAT HELPED ME RETRIEVE MY LOST INVESTMENT FUNDS : THE HACK ANGELS If you’re lost your Bitcoin to scammers, and you’re searching for trustworthy recovery services, getting in touch with THE HACK ANGELS is the only way to get your Bitcoin back. After losing my Bitcoin investments. I was determined to find a solution. I began by researching online and found several recovery services claiming to retrieve lost Bitcoin. However, most of these services turned out to be scams themselves. I thought I’d never see my funds again. It was then that I came across THE HACK ANGELS. A reputable Bitcoin recovery expert with a proven track record of retrieving lost Funds. I reached out to him, and he responded promptly, explaining the process of recovering my lost Bitcoin. They provide essential guidance on protecting digital assets, recognizing cyber threats and using effective security practices to prevent future incidents. Great communication and a successful outcome, when it comes to bitcoins recovery, quick contact THE HACK ANGELS RECOVERY EXPERT WhatsApp +1(520)2 0 0-2 3 2 0 Email at [email protected] Website at www.thehackangels.com 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. Don’t hesitate to reach out if you need help! And ready to reclaim what's rightfully yours?
11.08.25 10:33 Morris Jason

HIRE A REPUTABLE RECOVERY FIRM THAT HELPED ME RETRIEVE MY LOST INVESTMENT FUNDS : THE HACK ANGELS If you’re lost your Bitcoin to scammers, and you’re searching for trustworthy recovery services, getting in touch with THE HACK ANGELS is the only way to get your Bitcoin back. After losing my Bitcoin investments. I was determined to find a solution. I began by researching online and found several recovery services claiming to retrieve lost Bitcoin. However, most of these services turned out to be scams themselves. I thought I’d never see my funds again. It was then that I came across THE HACK ANGELS. A reputable Bitcoin recovery expert with a proven track record of retrieving lost Funds. I reached out to him, and he responded promptly, explaining the process of recovering my lost Bitcoin. They provide essential guidance on protecting digital assets, recognizing cyber threats and using effective security practices to prevent future incidents. Great communication and a successful outcome, when it comes to bitcoins recovery, quick contact THE HACK ANGELS RECOVERY EXPERT WhatsApp +1(520)2 0 0-2 3 2 0 Email at [email protected] Website at www.thehackangels.com 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. Don’t hesitate to reach out if you need help! And ready to reclaim what's rightfully yours?
11.08.25 10:34 Morris Jason

HIRE A REPUTABLE RECOVERY FIRM THAT HELPED ME RETRIEVE MY LOST INVESTMENT FUNDS : THE HACK ANGELS If you’re lost your Bitcoin to scammers, and you’re searching for trustworthy recovery services, getting in touch with THE HACK ANGELS is the only way to get your Bitcoin back. After losing my Bitcoin investments. I was determined to find a solution. I began by researching online and found several recovery services claiming to retrieve lost Bitcoin. However, most of these services turned out to be scams themselves. I thought I’d never see my funds again. It was then that I came across THE HACK ANGELS. A reputable Bitcoin recovery expert with a proven track record of retrieving lost Funds. I reached out to him, and he responded promptly, explaining the process of recovering my lost Bitcoin. They provide essential guidance on protecting digital assets, recognizing cyber threats and using effective security practices to prevent future incidents. Great communication and a successful outcome, when it comes to bitcoins recovery, quick contact THE HACK ANGELS RECOVERY EXPERT WhatsApp +1(520)2 0 0-2 3 2 0 Email at [email protected] Website at www.thehackangels.com 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. Don’t hesitate to reach out if you need help! And ready to reclaim what's rightfully yours?
11.08.25 10:34 Morris Jason

HIRE A REPUTABLE RECOVERY FIRM THAT HELPED ME RETRIEVE MY LOST INVESTMENT FUNDS : THE HACK ANGELS If you’re lost your Bitcoin to scammers, and you’re searching for trustworthy recovery services, getting in touch with THE HACK ANGELS is the only way to get your Bitcoin back. After losing my Bitcoin investments. I was determined to find a solution. I began by researching online and found several recovery services claiming to retrieve lost Bitcoin. However, most of these services turned out to be scams themselves. I thought I’d never see my funds again. It was then that I came across THE HACK ANGELS. A reputable Bitcoin recovery expert with a proven track record of retrieving lost Funds. I reached out to him, and he responded promptly, explaining the process of recovering my lost Bitcoin. They provide essential guidance on protecting digital assets, recognizing cyber threats and using effective security practices to prevent future incidents. Great communication and a successful outcome, when it comes to bitcoins recovery, quick contact THE HACK ANGELS RECOVERY EXPERT WhatsApp +1(520)2 0 0-2 3 2 0 Email at [email protected] Website at www.thehackangels.com 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. Don’t hesitate to reach out if you need help! And ready to reclaim what's rightfully yours?
11.08.25 11:35 Dennisblaq

Hi everyone, I'm an entrepreneur, and when I learned about cryptocurrencies and how they potentially triple my income, I was excited to invest. I hope this message finds the correct audience. On the internet, I came across someone who assured me of exceptionally significant returns on my investment. I was readily persuaded by this man's seeming wisdom. I quickly discovered that this cryptocurrency was a hoax and that I was unable to withdraw my money after making a substantial investment. My business was in tatters, and I owed money.Fortunately for me, a pop-up regarding cryptocurrency recovery and cybersecurity specialists that could help me get all of my money back appeared when seeking for help on the internet. This organization, WIZARD JAMES RECOVERY, is a cyber security and cryptocurrency recovery firm that helped me get my investment money back. I sincerely appreciate their assistance and heartily endorse them to anyone in need of cryptocurrency recovery. The following is their contact information: E - M A I L: [email protected] W H A T S A P P: (+ 44 741 836 7204)
11.08.25 13:05 alaincharlotte5

RECLAIM CRYPTO LOSS: HIRE META TECH RECOVERY PRO TO RECOVER LOST ASSET BACK. The adage "all that glitters is not gold" proved a harsh reality I recently encountered. The digital realm of cryptocurrency investments and exchange platforms is not immune to deception. My pursuit of cryptocurrency investments led me to a seemingly legitimate site, complete with convincing investor testimonials touting successful withdrawals. Unbeknownst to me, it was an elaborate scheme. I invested $210,000.00 worth of cryptocurrency across multiple wallets, as directed by the scammers. Initially, the process mirrored typical modern investments, but my withdrawal requests were met with evasiveness and silence. This represented my entire life savings, accumulated through years of dedicated work, now vanished in an instant. Refusing to accept defeat, I sought assistance from METATECH-RECOVERYPRO, which facilitated a successful recovery. Their process was remarkably efficient. They meticulously traced my lost funds back to the perpetrators, and within a short timeframe, the recovered assets were returned to my bank account. Without this exceptional recovery team, which offered support when all hope seemed lost, my situation would have been dire. To initiate your recovery, I strongly advise sending a detailed email ([email protected]) of your task, outlining your specific circumstances. Ask (META TECH RECOVERY PRO) for help via: Telegram:@metatechrecoveryproteam W/S +1 (469) 692‑8049 thank you.
11.08.25 13:05 alaincharlotte5

RECLAIM CRYPTO LOSS: HIRE META TECH RECOVERY PRO TO RECOVER LOST ASSET BACK. The adage "all that glitters is not gold" proved a harsh reality I recently encountered. The digital realm of cryptocurrency investments and exchange platforms is not immune to deception. My pursuit of cryptocurrency investments led me to a seemingly legitimate site, complete with convincing investor testimonials touting successful withdrawals. Unbeknownst to me, it was an elaborate scheme. I invested $210,000.00 worth of cryptocurrency across multiple wallets, as directed by the scammers. Initially, the process mirrored typical modern investments, but my withdrawal requests were met with evasiveness and silence. This represented my entire life savings, accumulated through years of dedicated work, now vanished in an instant. Refusing to accept defeat, I sought assistance from METATECH-RECOVERYPRO, which facilitated a successful recovery. Their process was remarkably efficient. They meticulously traced my lost funds back to the perpetrators, and within a short timeframe, the recovered assets were returned to my bank account. Without this exceptional recovery team, which offered support when all hope seemed lost, my situation would have been dire. To initiate your recovery, I strongly advise sending a detailed email ([email protected]) of your task, outlining your specific circumstances. Ask (META TECH RECOVERY PRO) for help via: Telegram:@metatechrecoveryproteam W/S +1 (469) 692‑8049 thank you.
11.08.25 13:05 alaincharlotte5

RECLAIM CRYPTO LOSS: HIRE META TECH RECOVERY PRO TO RECOVER LOST ASSET BACK. The adage "all that glitters is not gold" proved a harsh reality I recently encountered. The digital realm of cryptocurrency investments and exchange platforms is not immune to deception. My pursuit of cryptocurrency investments led me to a seemingly legitimate site, complete with convincing investor testimonials touting successful withdrawals. Unbeknownst to me, it was an elaborate scheme. I invested $210,000.00 worth of cryptocurrency across multiple wallets, as directed by the scammers. Initially, the process mirrored typical modern investments, but my withdrawal requests were met with evasiveness and silence. This represented my entire life savings, accumulated through years of dedicated work, now vanished in an instant. Refusing to accept defeat, I sought assistance from METATECH-RECOVERYPRO, which facilitated a successful recovery. Their process was remarkably efficient. They meticulously traced my lost funds back to the perpetrators, and within a short timeframe, the recovered assets were returned to my bank account. Without this exceptional recovery team, which offered support when all hope seemed lost, my situation would have been dire. To initiate your recovery, I strongly advise sending a detailed email ([email protected]) of your task, outlining your specific circumstances. Ask (META TECH RECOVERY PRO) for help via: Telegram:@metatechrecoveryproteam W/S +1 (469) 692‑8049 thank you.
11.08.25 17:49 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.08.25 17:49 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.08.25 17:49 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.08.25 19:46 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me $800,000 out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram 📱(@RoseCyberHives) and email rosecryptorecovery@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram 📱 (@RoseCyberHives) rosecryptorecovery@gmail. com
11.08.25 20:06 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me $800,000 out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram 📱(@RoseCyberHives) and email rosecryptorecovery@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram 📱 (@RoseCyberHives) rosecryptorecovery@gmail. com
12.08.25 02:43 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.08.25 02:43 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.08.25 02:43 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.08.25 16:48 ahmed11

It takes a team to recover lost money from scams. Recovery professionals locate stolen money by using their expertise. The best course of action is advised by legal professionals. This helps you get your money back. Identifying warning indicators is a crucial lesson learned from frauds. Promises of large, fast, risk-free profits should be avoided. They may put pressure on you to make an investment right away. For assistance avoiding frauds in the future, get in touch with Marie. She can also help you retrieve your misplaced Bitcoin. You can reach her on Telegram at @Marie_consultancy or via email at [email protected]. Also, you can use WhatsApp at +1 7127594675.
12.08.25 17:13 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
12.08.25 17:13 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
12.08.25 18:44 edinavida027

CRYPTO TRACING AND RECOVERY: HIRE A LEGITIMATE RECOVERY EXPERT" CONTACT RAPID DIGITAL RECOVERY
12.08.25 18:44 edinavida027

A recruiter named Emma Zhang reached out to me on LinkedIn regarding an intriguing job opportunity that piqued my interest. She presented a position for a Crypto Security Specialist with a Coinbase-certified firm. Her profile appeared legitimate, boasting over 1200 connections and numerous endorsements from industry professionals that seemed authentic. The offer was enticing: a salary of $180,000 plus bonuses, all disbursed in Bitcoin. The recruiter promptly sent me an email from a seemingly official domain , which closely resembled Coinbase's actual email addresses. It felt like an incredible opportunity. However, there was a stipulation: to qualify for the position, I needed to confirm that I possessed at least 4 BTC. This requirement struck me as peculiar but not entirely implausible, given the nature of the role. At that moment, I had sufficient BTC to meet the demand, so I decided to proceed. The HR representative directed me to a portal that closely mimicked Coinbase's official interface. It was convincing enough that I didn’t hesitate to log in and complete the necessary steps. Unfortunately, that was where everything began to unravel. I was instructed to send a verification deposit of 1 BTC to confirm my ownership. Since everything appeared legitimate, I complied without hesitation. However, the moment I transferred the BTC, an unsettling feeling washed over me. The following day, I noticed that the remaining 3 BTC in my wallet began to vanish. My heart sank as I realized something was terribly amiss. I immediately attempted to reach out to both the HR representative and the company, but my efforts were met with silence. Upon checking the LinkedIn profiles of Emma Zhang and her team, I discovered they had mysteriously vanished. I was left with nothing no job offer, no recruiter, and no Bitcoin. I turned to the internet in search of solutions. That’s when I stumbled upon RAPID DIGITAL RECOVERY, a company specializing in Bitcoin recovery. I contacted them, and they assured me they could assist in recovering my lost funds. After a tense and arduous process, they successfully retrieved my stolen BTC. The relief I felt was indescribable, and I am profoundly grateful to RAPID DIGITAL RECOVERY for their expertise and unwavering support. CONTACT INFO BELOW Email: rapid digital recovery (@) execs. com Telegram: htt ps: // t. me/ Rapid digital recovery519 WhatSapp: +1 4 1 4 8 0 7 1 4 8 5
12.08.25 19:15 vallatjosette

Fortunately, my luck changed when I encountered Wizard Hilton Cyber Tech. Their expertise and dedication to helping victims of cryptocurrency scams are truly remarkable. They worked tirelessly to track down the scammers and recover my lost crypto coins.Email : wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com
12.08.25 23:07 babatunde162

I recommend Marie when it comes to recovering lost/stolen usdt/bitcoin or any kind of cryptocurrencies' from fake investment platforms because they're well specialized in that area and you'll get your money back in full. I can boldly say this right now based on my prior deal i had with them; she was the only one who was able to recover my lost money $52,760 dollars back to my account, Only • ([email protected] and telegram:@Marie_consultancy or whatsapp +1 7127594675) she was successful in recovering my money. They are the only one who can fully restore your lost funds to your account without any deductions, I really value their work and am recommending her to you today. THANK ME LATER
13.08.25 06:11 tyler231

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13.08.25 07:01 Morris Jason

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13.08.25 07:01 Morris Jason

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13.08.25 07:01 Morris Jason

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13.08.25 08:26 Morris Jason

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13.08.25 08:27 Morris Jason

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13.08.25 08:27 Morris Jason

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13.08.25 10:37 babatunde162

Since Marie is an expert in recovering lost or stolen USDT, bitcoin, or any other cryptocurrency from fraudulent investment platforms, I suggest them for this task. You will receive your money back in full. I can confidently state this at this time because of my previous transaction with them: she was the only one who successfully returned $52,760 of my lost funds to my account. Her contact information is [email protected], and her telegram handle is @Marie_consultancy or +1 7127594675. Because they are the only ones that can completely return your lost money to your account without taking any money out, I truly appreciate what they do and am suggesting her to you today. Afterward, thank you.
13.08.25 13:15 Bartlevi

Sylvester showcased remarkable talent. He brought back over $610,000 in crypto. This felt like a massive win. His help went beyond this first success. He worked hard to get back more lost money. He stayed in touch all the time. He kept me in the loop. The whole process was very honest. There were no hidden fees or tricky ways. My money went straight to my crypto account. The transfer was easy. Sylvester Bryant is the most honest recovery agent I know. If you're in a similar spot, contact him now. You can email Sylvester at yt7cracker@gmail. com. Or reach him on WhatsApp at +1 (512) 577-7957. Don't give up hope. Sylvester can find your stolen money. He did it for me. You can get your money back. Sylvester Bryant will help you recover your funds.
13.08.25 14:23 Morris Jason

HIRE A REPUTABLE RECOVERY FIRM THAT HELPED ME RETRIEVE MY LOST INVESTMENT FUNDS : THE HACK ANGELS If you’re lost your Bitcoin to scammers, and you’re searching for trustworthy recovery services, getting in touch with THE HACK ANGELS is the only way to get your Bitcoin back. After losing my Bitcoin investments. I was determined to find a solution. I began by researching online and found several recovery services claiming to retrieve lost Bitcoin. However, most of these services turned out to be scams themselves. I thought I’d never see my funds again. It was then that I came across THE HACK ANGELS. A reputable Bitcoin recovery expert with a proven track record of retrieving lost Funds. I reached out to him, and he responded promptly, explaining the process of recovering my lost Bitcoin. They provide essential guidance on protecting digital assets, recognizing cyber threats and using effective security practices to prevent future incidents. Great communication and a successful outcome, when it comes to bitcoins recovery, quick contact THE HACK ANGELS RECOVERY EXPERT WhatsApp +1(520)2 0 0-2 3 2 0 Email at [email protected] Website at www.thehackangels.com 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. Don’t hesitate to reach out if you need help! And ready to reclaim what's rightfully yours?
13.08.25 14:23 Morris Jason

HIRE A REPUTABLE RECOVERY FIRM THAT HELPED ME RETRIEVE MY LOST INVESTMENT FUNDS : THE HACK ANGELS If you’re lost your Bitcoin to scammers, and you’re searching for trustworthy recovery services, getting in touch with THE HACK ANGELS is the only way to get your Bitcoin back. After losing my Bitcoin investments. I was determined to find a solution. I began by researching online and found several recovery services claiming to retrieve lost Bitcoin. However, most of these services turned out to be scams themselves. I thought I’d never see my funds again. It was then that I came across THE HACK ANGELS. A reputable Bitcoin recovery expert with a proven track record of retrieving lost Funds. I reached out to him, and he responded promptly, explaining the process of recovering my lost Bitcoin. They provide essential guidance on protecting digital assets, recognizing cyber threats and using effective security practices to prevent future incidents. Great communication and a successful outcome, when it comes to bitcoins recovery, quick contact THE HACK ANGELS RECOVERY EXPERT WhatsApp +1(520)2 0 0-2 3 2 0 Email at [email protected] Website at www.thehackangels.com 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. Don’t hesitate to reach out if you need help! And ready to reclaim what's rightfully yours?
13.08.25 19:46 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.08.25 19:46 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.08.25 19:46 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.08.25 19:50 babatunde162

When it comes to recovering lost or stolen bitcoin, USDT, or any other cryptocurrency from fraudulent investment platforms, I suggest Marie because they are highly skilled in the field and will return all of your money. Based on my previous transaction with them, I can state with confidence that she was the only one who successfully returned $52,760 of my lost funds to my account. Her email address is [email protected], and her telegram handle is @Marie_consultancy, or her WhatsApp number is +1 7127594675. I truly appreciate their work and am suggesting her to you today because they are the only ones who can completely return your missing money to your account without any deductions.
14.08.25 00:20 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
14.08.25 00:20 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
14.08.25 17:47 Joanne Ruth

The innovative company Captain Jack Recovery was created to address the challenges associated with recovering cryptocurrency assets. A group of seasoned professionals with backgrounds in offshore legal solutions, asset recovery, crypto intelligence, and crypto investigations founded Captain Jack Recovery. For people and companies who have been the victims of bitcoin fraud or theft, his goal is to offer complete recovery options. Conatc address ([email protected])
14.08.25 18:27 lisared

Losing nearly 500,000 USDT to hackers was one of the most devastating experience of my life. I felt powerless as my hard-earned assets vanished into anonymous blockchain transactions until I found Agent Jasmine Lopez. Her deep expertise, tireless effort, and unwavering focus gave me hope and delivered results. She traced complex transactions, pursued every lead, and recovered a large portion of my stolen funds something I never thought possible. If you’ve lost digital assets, don’t give up. Contact her at recoveryfundprovider [AT] gmail [DOT] com Whats//App at +{44 736-644-5035}. She turned my nightmare into a recovery story, and she could do the same for you
14.08.25 19:23 Joanne Ruth

The innovative company Captain Jack Recovery was created to address the challenges associated with recovering cryptocurrency assets. A group of seasoned professionals with backgrounds in offshore legal solutions, asset recovery, crypto intelligence, and crypto investigations founded Captain Jack Recovery. For people and companies who have been the victims of bitcoin fraud or theft, his goal is to offer complete recovery options. Conatc address ([email protected])
14.08.25 21:49 Rickbayford

If you need to recover cryptocurrency that has been lost to wallet hackers, internet scammers, or BTC transferred to the wrong address, I suggest Wizard James Recovery, a very trustworthy recovery organization. Just by sending my case to the Expert and supplying the necessary data, Wizard James Recovery was able to help me recover my Bitcoin. I'm glad I was able to recover this much after losing even more to the fraudulent broker I initially put my belief in. Errors are unavoidable, thus we can never be careful enough. If you have lost money to cryptocurrency scams, I strongly advise getting in touch with Wizard James Recovery. Mail: ([email protected])
15.08.25 03:10 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me $800,000 out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram (@RoseCyberHives) and email rosecryptorecovery@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram (@RoseCyberHives) rosecryptorecovery@gmail. com
15.08.25 04:59 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
15.08.25 04:59 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
15.08.25 04:59 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
15.08.25 04:59 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
15.08.25 09:13 eva117

Recovery experts and legal experts work together to get back stolen money. They use special skills to find and retrieve assets. This teamwork improves your chances of success. Learning to spot scam warnings is key. Look out for promises of big returns with no risk. Be wary of pressure to invest fast. Contact Marie at [email protected], Telegram @Marie_consultancy, or WhatsApp +1 7127594675. She can help you avoid future scams and recover lost bitcoin
15.08.25 13:27 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.08.25 13:27 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.08.25 20:26 caitlinfedex

After my wife passed, I found myself lonely and searching for connection. I joined a senior living forum just to talk with others my age to share stories, maybe feel a little less isolated. That’s where I met a man who claimed to be helping retirees like me manage their savings with cryptocurrency. He spoke warmly and convincingly about a retirement care fund that guaranteed lifetime monthly payouts. He said it was specifically designed for people in my stage of life safe secure and backed by health organizations. He gained my trust over weeks of conversation. We spoke daily. He called me by name asked about my late wife, and remembered little details about my life. I believed he genuinely cared. Eventually, I transferred $200000 in USDT the majority of my retirement savings, into the wallet address he provided. That was the last time I heard from him. The forum profile vanished. His phone number stopped working. No response to emails. I was stunned. I couldn’t believe I had been tricked. I didn’t know who to turn to. I felt ashamed and foolish. At my age recovering from a financial loss like this is nearly impossible.My relative stepped in. They searched online for solutions and found TRUST GEEKS HACK EXPERT. I was skeptical at first but they insisted we give them a chance. From the moment we contacted them they treated me with respect patience and genuine concern. Their team didn’t speak in technical riddles. They explained every step. They used blockchain analysis tools to trace the movement of my funds through several addresses.It turned out the wallet was linked to a scam platform. TRUST GEEKS HACK EXPERT worked directly with the USDT issuer and coordinated with relevant platforms to freeze the funds before they were fully laundered.In just two weeks they managed to recover $155000 in USDT.The money helped yes but it was more than that. I no longer felt powerless. I had been taken advantage of but I wasn’t alone in the fight. Thanks to TRUST GEEKS HACK EXPERT I got back not only a large part of my savings but also peace of mind a sense of dignity and hope for the future. If you ever find yourself in such situation I encourage you to contact TRUST GEEKS HACK EXPERT Email.trustgeekshackexpert{At}fastservice{Dot}com — Telegram:: Trustgeekshackexpert & W h a t's A p p +1 7 1 9 4 9 2 2 6 9 3
15.08.25 20:27 caitlinfedex

After my wife passed, I found myself lonely and searching for connection. I joined a senior living forum just to talk with others my age to share stories, maybe feel a little less isolated. That’s where I met a man who claimed to be helping retirees like me manage their savings with cryptocurrency. He spoke warmly and convincingly about a retirement care fund that guaranteed lifetime monthly payouts. He said it was specifically designed for people in my stage of life safe secure and backed by health organizations. He gained my trust over weeks of conversation. We spoke daily. He called me by name asked about my late wife, and remembered little details about my life. I believed he genuinely cared. Eventually, I transferred $200000 in USDT the majority of my retirement savings, into the wallet address he provided. That was the last time I heard from him. The forum profile vanished. His phone number stopped working. No response to emails. I was stunned. I couldn’t believe I had been tricked. I didn’t know who to turn to. I felt ashamed and foolish. At my age recovering from a financial loss like this is nearly impossible.My relative stepped in. They searched online for solutions and found TRUST GEEKS HACK EXPERT. I was skeptical at first but they insisted we give them a chance. From the moment we contacted them they treated me with respect patience and genuine concern. Their team didn’t speak in technical riddles. They explained every step. They used blockchain analysis tools to trace the movement of my funds through several addresses.It turned out the wallet was linked to a scam platform. TRUST GEEKS HACK EXPERT worked directly with the USDT issuer and coordinated with relevant platforms to freeze the funds before they were fully laundered.In just two weeks they managed to recover $155000 in USDT.The money helped yes but it was more than that. I no longer felt powerless. I had been taken advantage of but I wasn’t alone in the fight. Thanks to TRUST GEEKS HACK EXPERT I got back not only a large part of my savings but also peace of mind a sense of dignity and hope for the future. If you ever find yourself in such situation I encourage you to contact TRUST GEEKS HACK EXPERT Email.trustgeekshackexpert{At}fastservice{Dot}com — Telegram:: Trustgeekshackexpert & W h a t's A p p +1 7 1 9 4 9 2 2 6 9 3
15.08.25 21:04 vallatjosette

My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
16.08.25 06:53 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.08.25 06:53 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.08.25 06:53 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.08.25 07:01 lisared

Losing nearly 500,000 USDT to hackers was one of the most devastating experience of my life. I felt powerless as my hard-earned assets vanished into anonymous blockchain transactions until I found Agent Jasmine Lopez. Her deep expertise, tireless effort, and unwavering focus gave me hope and delivered results. She traced complex transactions, pursued every lead, and recovered a large portion of my stolen funds something I never thought possible. If you’ve lost digital assets, don’t give up. Contact Recoveryfundprovider@gmail. com WhatsApp at +{44 736-644-5035}. She turned my nightmare into a recovery story, and she could do the same for you.
16.08.25 17:05 mark

I see a lot of recommendations online and it’s already obvious there are bad eggs online who will only add to your mystery. I can only recommend one and you can reach them via mail on (zattechrecovery @ gmail c0 m ) if you need help on recovering what you lost to scammers.
17.08.25 03:33 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 03:33 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 05:22 aveasley4

I’m a California resident who was contacted by someone named “Lin” on WhatsApp. Our conversation eventually turned to cryptocurrency investments. Lin introduced me to a trading platform called CBOTBIT (https://cbotbit.com), walked me through setting up an account, and guided me on transferring funds. Under Lin’s direction, I began trading and was led to believe that my account was growing significantly. However, when I attempted to withdraw my funds, I was unable to do so — and shortly after, the website went offline entirely. I had invested over $340,000 in Bitcoin, planning to use it for my retirement. The experience was devastating, and I was at a loss for what to do next. Eventually, I came across [email protected] and decided to reach out. I followed their instructions, and within a few days, I was able to recover my original investment. I'm incredibly relieved and grateful for their assistance, and I recommend contacting them if you’re in a similar situation and need help recovering lost funds.
17.08.25 17:49 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 17:49 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 17:49 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.08.25 21:10 Connorhelms

I can honestly say losing my bitcoin savings was one of the darkest moments of my life. I had put everything I could into that investment, believing it would secure my future, only to wake up one day and realize the platform had vanished with all my funds. The feeling was indescribable, fear, shame, and helplessness all at once. I thought I’d never see that money again. Then I came across Alpha Spy Nest. At first, I was hesitant, I had already been deceived once, and the thought of trusting anyone else terrified me. But something about their professionalism and the way they patiently explained things gave me a little hope. For the first time in weeks, I felt like I wasn’t completely alone in the fight. They listened to my story without judgment, broke down the technical side in a way I could understand, and reassured me that there was still a chance. Watching them work was incredible. They traced every transaction, uncovered where my bitcoin had been moved, and kept me updated through every stage. It felt like they cared as much about my recovery as I did. The day they told me they had successfully recovered my lost savings, I actually cried. Seeing those funds back in my wallet felt unreal, it was like a huge weight had been lifted off my shoulders. Beyond just recovering my bitcoin, Alpha Spy Nest gave me back my peace of mind and restored a trust I thought I had lost forever. I’m forever grateful to them. If I hadn’t found Alpha Spy Nest, I’d still be stuck in regret and despair. Instead, I now share my story as proof that recovery is possible. You can also reach out to them via: Email: [email protected], WhatsApp: +15132924878‬, Telegram: https://t.me/Alphaspynest, Website: www.alphaspynest.org
18.08.25 22:17 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.08.25 22:17 wendytaylor015

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