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Машинное обучение: Кластеризация методом K-means. Теория и реализация. С нуля

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Здравствуйте, дорогие читатели. В этой статье я приведу разбор того, как работает метод кластеризации К-средних на низком уровне.

Содержание: идея метода, как присваивать метки неразмеченным объектам, реализация на чистом Python и разбор кода.

Введение

Кластеризация методом K-means направлена на разбиение набора данных на k кластеров, таких, что каждый объект принадлежит кластеру с ближайшим центром кластера. Цель алгоритма — минимизировать суммарное расстояние точек кластеров от их центров.

Кластеризацию можно рассматривать как классификацию без размеченных данных (без учителя), то есть алгоритм должен классифицировать данные сам, без маркировки. Ученик должен сам отнести объекты к определенным классам, но ученик не знает, какие это могут быть классы, поэтому нужно просто сказать ему, на сколько классов нужно разделить данные объекты (на k классов). Затем ученик начинает разделять объекты по их похожести, на k классов, то есть близко похожие друг на друга объекты ученик относит к одному классу.

Формулировка задачи

Пусть имеется набор данных, состоящий из n наблюдений (объектов) x_i = (x_{i1}, ..., x_{ik}), i=1,...,n, k - число признаков. На основе этих данных можно сформировать исходный набор объектов, на основе которого можно произвести кластеризацию этих объектов, то есть найти центры кластеров — центры скоплений объектов разных классов.

Идея метода K-means. Необходимо определить количество кластеров k, которое нужно выделить в данных. Это значение обычно задается заранее, хотя существуют методы для его определения.

Выбираются k случайных центров кластеров для набора данных. Эти центры могут быть выбраны случайным образом или используя специальные методы инициализации.

Каждая точка данных присваивается к ближайшему центру кластера. Это делается путем расчета расстояния (обычно евклидово) между каждой точкой данных и каждым центром.

Центры обновляются как среднее значение всех точек, принадлежащих каждому кластеру. Этот шаг повторяется до тех пор, пока центры не перестанут меняться или до достижения заданного критерия сходимости. Критерием сходимости может быть качество модели либо разница изменения центров кластеров между предыдущей и текущей итерацией.

Суть метода в том, чтобы сначала найти центры скоплений объектов, а потом, на основе этих центров, классифицировать объекты путем присвоения каждому объекту в исходном наборе данных принадлежность к определенному классу (кластеру).

Разделение точек данных на кластеры

Нужно определить для каждой точки в исходном наборе данных к какому кластеру (классу) она относится. Моделью KNN будут являться метрики, алгоритмы, используемые в определении класса точек и присвоенные всем точкам их метки.

Задаем число кластеров. Возможно это можно сделать динамически, но для этого нужно придумать алгоритм, который будет делить данные на кластеры основываясь на каких-то общих закономерностях в данных, а не просто на основе расстояния.

Потом создаем центры кластеров, например, случайно. Можно задать центры кластеров по графику, на графике видно, в каких областях происходят скопления объектов, центрами кластеров будут центры этих областей.

Далее для каждой рассматриваемой точки нужно вычислить расстояние между рассматриваемой точкой данных и всеми центрами кластеров. Рассматриваем каждую точку в наборе данных, вычисляем расстояния до всех точек-центров кластеров, выбираем кластер, до которого наименьшее расстояние.

Евклидово расстояние между двумя точками вычисляется по этой формуле:

d(a, b) = \sqrt{\sum_{i = 1}^{n} (a_i - b_i)^2}

a и b — точки данных (объекты, векторы признаков), a_i и b_i - элементы векторов (признаки), n — количество признаков в векторе (количество координат).

Действие алгоритма таково, что он стремится минимизировать сумму квадратов отклонений точек кластеров от центров этих кластеров:

S = \sum_{i=1}^{n} \sum_{j = 1}^{p} (x_{ij} - c_{kj})^2

n — число точек в одном кластере, p — число элементов вектора (число признаков, координат), c — центр k-го кластера, x_i - i-я точка в кластере, x_{ij} - j-я координата вектора x_i. Эта формула по сути и есть критерий сходимости алгоритма. Если значение этой функции станет меньше заданного, то алгоритм прекращает изменять центры кластеров и получается готовая модель К-средних.

Красные точки — случайные центры кластеров, черные — точки данных
Красные точки — случайные центры кластеров, черные — точки данных

Нужно просматривать каждую точку и присваивать ей кластер, к которому она находится ближе всего, для этого нужно вычислить расстояние до каждого центра кластера. Кластеры — это красные точки на рисунке, черные точки — объекты набора данных. Каждому объекту присваивается ближайший кластер (метка ближайшей красной точки).

После каждой итерации присвоения каждой точке определенной метки, нужно обновить значения центров кластеров. Чтобы это сделать, нужно вычислить для каждого кластера среднее значение всех точек, которые относятся к рассматриваемому кластеру. Суммируем все точки по каждой координате по отдельности, а потом делим на общее число точек:

C_{new, j} = \frac{1}{n} \sum_{i = 1}^{n} x_{ij}, j = 1,...,p

p — число координат, C — центр кластера, j — номер координаты (признака) объекта, n — число точек в кластере C, x_{ij} - j-я координата i-й точки кластера C. Нужно посчитать среднее значение для каждой координаты j.

Ограничения и преимущества

Алгоритм K-means не гарантирует достижения глобального минимума суммарного квадратичного отклонения. Он может застрять в локальном минимуме, зависящем от начальной инициализации центров. Поэтому можно пробовать несколько раз инициализировать центры заново.

Необходимо заранее определить количество кластеров k, которое нужно найти в данных. Для этого можно использовать дополнительные методы для определения числа кластеров.

Результат алгоритма зависит от начальной инициализации центров. Разные начальные центры могут привести к разным кластерам.

Алгоритм K-means относительно прост в реализации и понимании.

K-means может работать с огромными наборами данных и быстро обучается на новых примерах, он может быть использован для решения многих сложных задач машинного обучения.

Алгоритм K-means реализован в различных фреймворках машинного обучения.

Метод K-means является мощным инструментом для кластеризации данных, но требует тщательного выбора параметров и предварительной обработки данных для достижения оптимальных результатов.

Создание набора данных

Для задачи кластеризации нужны неразмеченные однотипные данные. Для этого используем функцию make_blobs из sklearn. Указываем число точек данных, количество центров точек (центров кластеров), число признаков (координат) и число для инициализации генератора псевдослучайных чисел.

from sklearn.datasets import make_blobs
from matplotlib import pyplot as plt

data, labels = make_blobs(n_samples=20, centers=2, n_features=2, random_state=0)
data = data.round(2)

print(list(labels))
points = [tuple(point) for point in data]
print(points)

x1_list = data[labels==0, 0]
x2_list = data[labels==1, 0]
y1_list = data[labels==0, 1]
y2_list = data[labels==1, 1]

plt.scatter(x=x1_list, y=y1_list, color='red')
plt.scatter(x=x2_list, y=y2_list, color='green')
plt.show()

Получаем двумерный массив точек и одномерный массив меток — к какому классу относится каждая точка (0 или 1 для двух центров). Делаем из массива список точек и выводим. Копируем эти данные в другой файл.

Набор данных. Сверку список меток, снизу — список точек
Набор данных. Сверку список меток, снизу — список точек

Далее выделяем из данных массив для каждого признака. Массив data — двумерный первая ось массива обозначает количество точек, сами точки (объекты), вторая ось массива обозначает значения элементов точек, то есть значения координат точек, выбираем определенные значения в массиве. В качестве первого индекса указываем те элементы, которые относятся к определенному классу (к 0 или к 1), в качестве второго индекса выбираем элементы массива, которые относятся к 0 или к 1 признаку (к координате x или к координате y).

Теперь рисуем точки на графике.

Размеченные точки
Размеченные точки
Неразмеченные точки
Неразмеченные точки

На графике видно 2 области скопления (красным и зеленым, рисунок сверху). Можно выделить два центра кластеров с примерными центрами, первый с координатами (1, 5), второй — (2.5, 1). Но если бы мы не знали, сколько всего должно быть кластеров (рисунок снизу), то можно было бы предположить, что их 3, 4 или больше. Но здесь мало объектов, чем больше объектов, тем было бы легче предположить число кластеров и их центры.

Пробная реализация

Делаем набор данных и задаем количество центров k и точки центров кластеров. Число кластеров в этом примере равно 2, а значения центров выбраны по графику.

target_labels = [0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0]
points = [(1.84, 3.56), (1.01, -0.52), (1.29, 3.45), (1.93, 4.15),
          (2.84, 3.33), (1.42, 4.64), (0.87, 4.71), (2.21, 1.28),
          (4.33, -0.56), (-1.58, 4.96), (2.1, 0.71), (1.17, -1.08),
          (3.59, 2.37), (1.12, 5.76), (1.67, 0.6), (3.29, 2.1),
          (1.71, 1.05), (0.35, 2.85), (2.47, 4.1), (1.74, 4.43)]

centers = [(1, 5), (2.5, 1)]  # центры кластеров, они выбраны вручную по графику

Делаем функцию для измерения квадратичного расстояния до каждого центра от рассматриваемой точки.

def euclidean_distances_to_centers(point):
    dists = []

    for center in centers:
        distance = 0

        for i in range(2):
            distance += (point[i] - center[i]) ** 2

        dists.append(distance)

    return dists

Проходимся по каждому центру и по каждой координате, записываем значения расстояний в список dists.

Далее делаем функцию для нахождения наименьшего расстояния, то есть наименьшего элемента в списке и возвращаем его индекс.

def get_min_item_idx(distances):  # найти минимальный элемент
    min_dist, min_item_idx = 1000000, 0

    for idx, dist in enumerate(distances):
        if dist < min_dist:
            min_dist = dist
            min_item_idx = idx

    return min_item_idx

Далее делаем функцию для обновления центров кластеров. Для этого понадобится записывать принадлежность точек к кластерам, записываем это в список clusters.

clusters = [[], []]

def update_centers():
    for cluster_idx, cluster in enumerate(clusters):
        cluster_mean_value = [0, 0]

        for point in cluster:
            for i, feature in enumerate(point):
                cluster_mean_value[i] += feature

        for i in range(2):
            cluster_mean_value[i] = cluster_mean_value[i] / len(cluster)

        centers[cluster_idx] = cluster_mean_value

Проходимся по каждому кластеру (списку точек), для каждого кластера задаем новое значение центра в переменную cluster_mean_value. Проходимся по каждой точке и по каждой координате точки, суммируем каждую координату у каждой точки. Потом делим получившиеся суммы на число точек в каждом кластере. Таким образом получается среднее значение всех точек в кластере — это и есть новое значение центра кластера.

Далее делаем цикл для итеративного изменения центров.

labels = []

for _ in range(100):
    clusters = [[], []]
    labels = []

    for point in points:
        dists = euclidean_distances_to_centers(point)
        min_item_idx = get_min_item_idx(dists)  # получить индекс класрера
        clusters[min_item_idx].append(point)  # добавляем точку в ближайший кластер
        labels.append(min_item_idx)

    update_centers()

print(labels)
output:
[0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0]

В каждой итерации присваиваем каждой точке принадлежность к определенному кластеру заново, записываем метки классов для каждой точки, потом, на основе нового присвоения, обновляем центры. Смотрим какие метки получились.

Это в принципе все, что нужно для этого метода. Далее можно посмотреть на графике, какое получилось разделение.

import numpy as np
from matplotlib import pyplot as plt

data = np.array(points)
labels = np.array(labels)

x1_list = data[labels==0, 0]
x2_list = data[labels==1, 0]
y1_list = data[labels==0, 1]
y2_list = data[labels==1, 1]

plt.scatter(x=x1_list, y=y1_list, color='red')
plt.scatter(x=x2_list, y=y2_list, color='green')
plt.show()

Делаем обратную процедуру, которую делали при создании набора данных, получилось почти то же самое.

Кластеризованные точки данных
Кластеризованные точки данных

Улучшение кода

Создадим класс для модели К-средних. Перед этим переделаем функцию для измерения расстояния.

def euclidean(point1, point2):
    return sum([(point1[i] - point2[i]) ** 2 for i in range(len(point1))])


def euclidean_distances(cur_point, points):
    dists = []

    for point in points:
        dists.append(euclidean(cur_point, point))

    return dists

Определяем класс. Для инициализации понадобится указать только число признаков.

class KMeans:
    def __init__(self, features_num):
        self.features_num = features_num
        self.centers = []
        self.clusters = [[] for _ in range(features_num)]

Далее делаем 2 метода для того, чтобы задавать центры.

def set_centers(self, centers):
    self.centers = centers

def init_centers(self, points, k):
    min_value = np.min(points, axis=0) * 100
    max_value = np.max(points, axis=0) * 100
    self.centers = []

    for _ in range(k):
        x = randint(min_value[0], max_value[0]) / 100
        y = randint(min_value[1], max_value[1]) / 100
        self.centers.append([x, y])

Центры можно будет задавать случайно или передавать заранее созданные. Случайные создаются на основе минимальных и максимальных значений признаков в точках данных. Предварительно нужно импортировать эти модули.

import numpy as np
from random import randint

Далее метод для обновления центров. Тут ничего не изменилось осталось так же, как и было.

def _update_centers_(self):
    for cluster_idx, cluster in enumerate(self.clusters):
        cluster_mean_value = [0, 0]
    
        for point in cluster:
            for i, feature in enumerate(point):
                cluster_mean_value[i] += feature
    
        for i in range(2):
            cluster_mean_value[i] = cluster_mean_value[i] / len(cluster)
    
        self.centers[cluster_idx] = cluster_mean_value

Функция для поиска индекса минимального элемента остается почти той же. Ее можно улучшить убрав переменную min_dist и использовать сравнение по индексу наименьшего элемента, вместо того, чтобы хранить значение этого элемента.

# найти индекс минимального элемента
def get_min_item_idx(distances):
    min_item_idx = 0

    for idx, dist in enumerate(distances):
        if dist < distances[min_item_idx]:
            min_item_idx = idx

    return min_item_idx

Теперь метод fit, в котором будет происходить итеративное изменение центров.

def fit(self, points, epochs):
    labels = []

    for _ in range(epochs):
        self.clusters = [[] for _ in range(self.features_num)]
        labels = []

        for point in points:
            dists = euclidean_distances(point, self.centers)
            min_item_idx = get_min_item_idx(dists)  # получить индекс класрера
            self.clusters[min_item_idx].append(point)
            labels.append(min_item_idx)

        self._update_centers_()

    return self.clusters, labels

Тут тоже ничего не изменилось. Добавляем возврат списков кластеров и меток из метода.

Теперь проверяем и тестируем.

kmeans = KMeans(features_num=2)
kmeans.set_centers(centers)
clusters, labels = kmeans.fit(points, 10)

print(labels)
output:
[0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0]

Заключение

Мы немного разобрали теорию кластеризации — рассмотрели суть метода К-средних и сделали реализацию модели. Сделали набор данных для задачи кластеризации с двумя признаками, сделали простую реализацию модели с нуля на чистом python, а потом улучшили код, сделали его более удобным для использования.

Такой обучающий урок будет полезен для начинающих и для тех, кому интересно разбираться во всем с нуля. Такой формат обучающих материалов показывает как происходит процесс разработки сложных решений от простых попыток реализации к полноценной рабочей системе, которую можно легко внедрить в какие-то программы.

Ссылка на папку с кодом из статьи на ГитХаб.

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  • 09.10.25 11:05 marcushenderson624

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  • 09.10.25 11:05 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

  • 09.10.25 11:05 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

  • 09.10.25 11:05 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

  • 11.10.25 04:41 luciajessy3

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  • 11.10.25 10:44 Tonerdomark

    A thief took my Dogecoin and wrecked my life. Then Mr. Sylvester stepped in and changed everything. He got back €211,000 for me, every single cent of my gains. His calm confidence and strong tech skills rebuilt my trust. Thanks to him, I recovered my cash with no issues. After months of stress, I felt huge relief. I had full faith in him. If a scam stole your money, reach out to him today at { yt7cracker@gmail . com } His help sparked my full turnaround.

  • 12.10.25 01:12 harristhomas7376

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

  • 12.10.25 01:12 harristhomas7376

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

  • 12.10.25 19:53 Tonerdomark

    A crook swiped my Dogecoin. It ruined my whole world. Then Mr. Sylvester showed up. He fixed it all. He pulled back €211,000 for me. Not one cent missing from my profits. His steady cool and sharp tech know-how won back my trust. I got my money smooth and sound. After endless worry, relief hit me hard. I trusted him completely. Lost cash to a scam? Hit him up now at { yt7cracker@gmail . com }. His aid turned my life around. WhatsApp at +1 512 577 7957.

  • 12.10.25 21:36 blessing

    Writing this review is a joy. Marie has provided excellent service ever since I started working with her in early 2018. I was worried I wouldn't be able to get my coins back after they were stolen by hackers. I had no idea where to begin, therefore it was a nightmare for me. However, things became easier for me after my friend sent me to [email protected] and +1 7127594675 on WhatsApp. I'm happy that she was able to retrieve my bitcoin so that I could resume trading.

  • 13.10.25 01:11 elizabethrush89

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

  • 13.10.25 01:11 elizabethrush89

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

  • 14.10.25 01:15 tyleradams

    Hi. Please be wise, do not make the same mistake I had made in the past, I was a victim of bitcoin scam, I saw a glamorous review showering praises and marketing an investment firm, I reached out to them on what their contracts are, and I invested $28,000, which I was promised to get my first 15% profit in weeks, when it’s time to get my profits, I got to know the company was bogus, they kept asking me to invest more and I ran out of patience then requested to have my money back, they refused to answer nor refund my funds, not until a friend of mine introduced me to the NVIDIA TECH HACKERS, so I reached out and after tabling my complaints, they were swift to action and within 36 hours I got back my funds with the due profit. I couldn’t contain the joy in me. I urge you guys to reach out to NVIDIA TECH HACKERS on their email: [email protected]

  • 14.10.25 08:46 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

  • 14.10.25 08:46 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

  • 14.10.25 08:46 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

  • 15.10.25 18:07 crypto

    Cryptocurrency's digital realm presents many opportunities, but it also conceals complex frauds. It is quite painful to lose your cryptocurrency to scam. You can feel harassed and lost as a result. If you have been the victim of a cryptocurrency scam, this guide explains what to do ASAP. Following these procedures will help you avoid further issues or get your money back. Communication with Marie ([email protected] and WhatsApp: +1 7127594675) can make all the difference.

  • 15.10.25 21:52 harristhomas7376

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

  • 15.10.25 21:52 harristhomas7376

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

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