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Матричные факторизации

Мы рассматриваем задачи рекомендаций. Нужно порекомендовать пользователю что-нибудь: статьи, фильмы, сериалы, музыку, видео, картинки, товары, и т.п.

В целом задачи рекомендаций можно представить как описание что произошло с пользователем и как ему надо провзаимодействовать. Описание пользователя можно представить по-разному:

  1. как неупорядоченный набор айтемов с которыми пользователь провзаимодействовал (список пролайканых треков)

  2. как неупорядоченный набор айтемов с оценками (оценка пользователем фильмов, или какой-то другой агрегат, например, сколько раз пользователь прослушал трек)

  3. как последовательность айтемов с оценками

  4. как последовательность взаимодействий с учетом контекста

Фитбек пользователя бывает:

  • explicit или явным (оценки, лайки)

  • implicit или неявным (клики, скипы)

Мы будем представлять пользователя как неупорядоченный набор айтемов с оценками и пытаться предсказать оценку пользователя на других айтемах.

И важно иметь ввиду, что матрица оценок R (таблица, в которой каждая строка представляет собой пользователя, каждый столбец - товар) - очень разреженная матрица.

Простейшими решениями являются Item- и User-based подходы.

Item- и User-based подходы (коллаборативная фильтрация (CF))

  • User2User - ищем похожесть между юзерами (топ ближайших пользователей, усредненные фитбеки разных пользователей)

  • Item2Item - ищем похожести между айтемами (например, средняя оценка по наиболее похожим фильмам)

Ниже приведены формулы для Item2Item, но, с точность до замен переменных, будет User2User.

Пусть s(i, j) - матрица похожестей (насколько айтемы i и j похожи).

В случае оценок (explicit фитбек) в качестве похожести можно считать корреляцию:

corr(i, j) = \frac{ \sum_u (r_{ui} - \overline r_{i}) (r_{uj} - \overline r_{j})}{ \sqrt{\sum_u (r_{ui} - \overline r_{i})^2 (r_{uj} - \overline r_{j})^2}}

В случае взаимодействий (implicit фитбек) можно посчитать меру Жаккара:

J(A, B) = \frac{|A \cup B|}{|A \cap B|} = \frac{|A \cup B|}{|A| + |B| - |A \cup B|}

Посчитав похожести айтемов (получится матрица Item \times Item) предсказание для пользователя для i-го айтема:

\hat r_{u i} = \frac{\sum_j s(i, j) (r_{uj} - r_{u})}{\sum_j |s(i, j)|} + r_u

На самом деле, в качестве элементов матрицы похожестей можно брать не только корреляцию.

Задача SLIMa (Sparse Linear Methods for Top-N Recommender Systems) как раз в том чтобы выучить матрицу похожестей (подобрать ее).

Считать в больших задачах корреляцию слишком тяжело. Обычно, в качестве похожести в каждой строчке для Item2Item оставляют самый ТОП. То есть у каждого айтема есть какое-то топовое количество айтемов на которые он похож (с весами похожести или без). Тогда текущий айтем будет просто усреднением ТОПа. Что, в каком-то смысле, похоже на KNN.

Выбор между Item2Item и User2User зависит от количества оценок на пользователя и на айтем - чем больше, тем надежнее похожесть. Это же приводит к возможности предрасчета: если оценок много, то добавление пары оценок ничего не изменит. Тогда можно обновляться раз в какое-то время. Сами похожести можно использовать для других задач (кандидат-генерация, контекстные рекомендации).

В продакшене в качестве чистого алгоритма рекомендаций ни Item2Item, ни User2User сегодня не используют. Однако, часто используют как какая-то часть рекомендаций.

PCA и SVD, Truncated SVD

Задача PSA - понизить размерность точек в многомерном пространстве линейным способом. Выбрать линейное подпространство L_k такое, чтобы сумма расстояний до него была минимальной (они наилучшим образом приближались этой линейной плоскостью).

Цель - сопоставить каждой точке проекцию на L_k.

\sum dist(x_i, L_k) \rightarrow min

Какая же связь у PSA с матричным разложением

Исходная матрица Rпоучила новые координаты в новом пространстве L_k. Пространство L_k имело какие-то базисные вектора (L_k матрица базиса или поворота). Если мы их запишем, они будут в матричке V.

R_{M \times N} \approx U_{M \times k} \times V^T_{k \times N} + B_{1 \times N}

Таким образом, мы пытаемся представить исходную марицу R в виде произведения двух матриц и добавить какое-то смещение:

r_{ij} = \overline u_i^T \overline v_j + b_i

Что же представляет из себя SVD

SVD раскладывает исходную матрицу в виде произведения трех матриц:

R = U \Sigma V^T\\U U^T = U^T U = I_M\\V V^T = V^T V = I_N

причем, U и V - матрицы поворота.

Если мы будем умножать вектор на SVD-разложение, то сначала мы вектор повернем, потом растянем и еще раз повернем.

При этом, в матрице \Sigma оставляют столько, сколько было не нулевых компонентов (rank(R)).

R = U \Sigma V^T \\l = rank(R) \\U^T U = V^T V = I_l

В результате получим усеченные матрицы поворота.

А отбросив самые малозначимые сингулярные числа в матрице \Sigma мы получим самое лучшее приближение для исходной матрицы R.

Truncated SVD как задача оптимизации:

R \approx U \Sigma V^T \\ U^T T = V^T V = I_k \\\sum_{\forall i, j} \big( r_{ij} - u_i^T \Sigma v_j \big)^2 \rightarrow \min_{U, V, \Sigma}U \Sigma V^T = X Y^T \\ X^T X \not = I_k, Y^T Y \not = I_k

Эквивалентная задача:

\sum_{\forall i, j} \big( r_{ij} - x_i^T y_j \big)^2 \rightarrow \min_{X, Y}\min_{X, Y} \sum_{\forall i, j} \big( r_{ij} - x_i^T y_j \big)^2 + \lambda \sum_{i} \|x_i\|^2 C_i + \lambda \sum_{j} \|y_j\|^2 C_j

где x_i называем вектором пользователя, а y_j - вектором айтема, C_i - коэффициент взаимодействия.

C_i = \frac{ \big| \{ j|r_{ij} > 0 \} \big| ^{\alpha} \big|\{i\}\big| }{\sum_i \big| \{ j|r_{ij} > 0 \} \big| ^{\alpha}}

Такая задача оптимизации решается градиентным спуском и довольно хорошо сходится.

Funk SVD: Восстанавливаем матрицу R только по заданным элементам.

Интерпретация SVD

Если вернуться к исходной постановке с SVD и развернуть матрицы X и Y правильно, чтобы они стали ортогональными, а нормы вынести в отдельную матрицу \Sigma между ними, то у этих векторов неожиданно будет появляться какой-то геометрический смысл. Часто можно подобрать какую-то интерпретацию (например, если речь про фильмы, то это могут быть популярность, комедия, ужастик и тд.).

Похожесть между айтемами измеряем как сosine similarity. Но результате, мы выдаем новые айтемы для пользователя, которые дают большое скалярное произведение с вектором пользователя. Чаще всего это будут айтемы с большой нормой. Но, регуляризация, в том числе, борется и с этим.

Алгоритм метода чередующихся наименьших квадратов (Alternating Least-Squares, ALS):

  • Инициализируем X и Y случайными значениями

  • В цикле:

    • Фиксируем матрицу X (пользователей).

    • Находим оптимальную матрицу Y (решаем гребневую регрессию для каждого товара).

    • И наоборот

ALS - шаг по x_i

Задача предствляет собой квадратичный функционал. Положим, если мы знаем вектор айтема, то получим квадратичную задачу по вектору юзера.

Минимизируя вектор айтема по оптимальному пользовательскому вектору, очевидно, что не нужно оптимизировать сумму квадратов айтемных векторов (они уже константно заданы).

\require{cancel} \min_{X, Y} \sum_{\forall i, j} \big( r_{ij} - x_i^T y_j \big)^2 + \lambda \sum_{i} \|x_i\|^2 C_i + \cancel{ \lambda \sum_{j} \|y_j\|^2 C_j }

Раскроем скобки. Уберем сумму квадратов элементов матрицы R, которая ни на что не влияет.

\arg \min_{x_i} \cancel{ \sum r_{ij}^2 } - 2 \sum r_{ij} x_i^T y_j + \sum (x_i^T y_j)^2 + \lambda \langle x_i, x_i \rangle C_i

Еще немного преобразуем (красным выделила места преобразований):

\arg \min_{x_i} -2 \color{red}{ x_i^T }\sum r_{ij} y_j + \sum \color{red}{ x_i^T y_j y_j^T x_i } + \lambda C_i \color{red}{ x_i^T x_i }\arg \min_{x_i} -2 \color{red}{ x_i^T } \big( \sum r_{ij} y_j \big) + \color{red}{ x_i^T } \big( \sum y_j y_j^T + \lambda C_i \big) \color{red}{ x_i }

Нужно получить оптимальное решение для:

\arg \min_{x_i} -2 x_i^T B_i + x_i^T A_i x_i

Матрично продифференцируем и прировняем к нулю:

-2B_i + 2A_ix = 0 \\ x_i = A_i^{-1} B_i

Таким образом оптимальная точка:

\arg \min_{x_i} -2 x_i^T B_i + x_i^T A_i x_i = A_i^{-1} B_i = \big( \sum y_j y_j^T + \lambda C_i \big) ^{-1} \big( \sum r_{ij} y_j \big)

IALS немного изменяет функционал оптимизации, что позволяет учитывать места, которые важно хорошо оптимизировать (было много взаимодействий).

Итоги SVD

  • Матричные факторизации (MF) не считывают похожести товаров напрямую (как коллаборативная фильтрация (CF))

  • MF пытаются описать пользователей и товары маленьким набором характеристик (часто меньше 100), объясняющих оценки

  • Эти характеристики могут быть неинтерпретируемыми. Как минимум, векторы айтемов и юзеров определены с точностью до поворота

  • SVD сложно распараллелить

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    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

    Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / WHATSAPP ; +1 (603) 702 ( 4335 ) is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…

  • 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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