The paper presents an efficient algorithm for reconstructing a low-rank matrix from a small number of observed entries. The algorithm, called Spectral Matrix Completion, is designed to handle large datasets where the number of entries \( n \) and the rank \( r \) are both large. The main result shows that the algorithm achieves a relative root mean square error (RMSE) of \( C(\alpha) (\frac{nr}{|E|})^{1/2} \) with \( |E| = O(rn) \) observed entries, where \( \alpha = m/n \). For \( r = O(1) \) and a sufficiently unstructured matrix, the algorithm can reconstruct the matrix exactly with \( |E| = O(n \log n) \) entries. The complexity of the algorithm is \( O(|E|r \log n) \), making it suitable for massive datasets. The proof of the main result involves a generalization of a result by Friedman-Kahn-Szemerédi and Feige-Ofek on the spectrum of sparse random matrices. The paper also discusses related work and open problems, including the optimal RMSE with \( O(n) \) entries and the threshold for exact completion.The paper presents an efficient algorithm for reconstructing a low-rank matrix from a small number of observed entries. The algorithm, called Spectral Matrix Completion, is designed to handle large datasets where the number of entries \( n \) and the rank \( r \) are both large. The main result shows that the algorithm achieves a relative root mean square error (RMSE) of \( C(\alpha) (\frac{nr}{|E|})^{1/2} \) with \( |E| = O(rn) \) observed entries, where \( \alpha = m/n \). For \( r = O(1) \) and a sufficiently unstructured matrix, the algorithm can reconstruct the matrix exactly with \( |E| = O(n \log n) \) entries. The complexity of the algorithm is \( O(|E|r \log n) \), making it suitable for massive datasets. The proof of the main result involves a generalization of a result by Friedman-Kahn-Szemerédi and Feige-Ofek on the spectrum of sparse random matrices. The paper also discusses related work and open problems, including the optimal RMSE with \( O(n) \) entries and the threshold for exact completion.