**Summary:** This paper presents an efficient algorithm for matrix completion from a few entries. The matrix $ M $ is assumed to have rank $ r \ll n $, and a subset $ E $ of its entries is observed. The algorithm reconstructs $ M $ from $ |E| = O(rn) $ observed entries with relative root mean square error (RMSE) bounded by $ C(\alpha)\left(\frac{nr}{|E|}\right)^{1/2} $. For sufficiently unstructured matrices with $ r = O(1) $, exact reconstruction is possible from $ |E| = O(n \log n) $ entries. The algorithm involves three steps: trimming the matrix to remove high-degree rows and columns, projecting the trimmed matrix to a rank-$ r $ matrix, and cleaning residual errors by minimizing a discrepancy function. The trimming step significantly improves the visibility of the underlying rank structure, especially when the number of revealed entries per row/column follows a heavy-tailed distribution. Theoretical analysis shows that the algorithm achieves arbitrarily small RMSE from $ O(nr) $ entries, and under certain incoherence conditions, it can reconstruct the matrix exactly from $ |E| \geq C'n r \sqrt{\alpha} (\Sigma_{\max}/\Sigma_{\min})^2 \max\{\log n, r\} $ entries. The complexity of the algorithm is $ O(|E|r \log n) $, making it suitable for large datasets. The paper also generalizes results from random matrix theory and provides a rigorous analysis of the algorithm's performance. It improves upon previous guarantees for matrix completion and demonstrates that the algorithm can handle real-world data with high accuracy. The results are validated through theoretical proofs and numerical simulations, showing that the algorithm performs well even for large matrices.**Summary:** This paper presents an efficient algorithm for matrix completion from a few entries. The matrix $ M $ is assumed to have rank $ r \ll n $, and a subset $ E $ of its entries is observed. The algorithm reconstructs $ M $ from $ |E| = O(rn) $ observed entries with relative root mean square error (RMSE) bounded by $ C(\alpha)\left(\frac{nr}{|E|}\right)^{1/2} $. For sufficiently unstructured matrices with $ r = O(1) $, exact reconstruction is possible from $ |E| = O(n \log n) $ entries. The algorithm involves three steps: trimming the matrix to remove high-degree rows and columns, projecting the trimmed matrix to a rank-$ r $ matrix, and cleaning residual errors by minimizing a discrepancy function. The trimming step significantly improves the visibility of the underlying rank structure, especially when the number of revealed entries per row/column follows a heavy-tailed distribution. Theoretical analysis shows that the algorithm achieves arbitrarily small RMSE from $ O(nr) $ entries, and under certain incoherence conditions, it can reconstruct the matrix exactly from $ |E| \geq C'n r \sqrt{\alpha} (\Sigma_{\max}/\Sigma_{\min})^2 \max\{\log n, r\} $ entries. The complexity of the algorithm is $ O(|E|r \log n) $, making it suitable for large datasets. The paper also generalizes results from random matrix theory and provides a rigorous analysis of the algorithm's performance. It improves upon previous guarantees for matrix completion and demonstrates that the algorithm can handle real-world data with high accuracy. The results are validated through theoretical proofs and numerical simulations, showing that the algorithm performs well even for large matrices.