This paper presents a comprehensive survey of matrix completion, a problem of recovering a low-rank matrix from a small number of its entries, which may be corrupted by noise. The authors show that under suitable conditions, one can recover a low-rank matrix from a nearly minimal set of entries by solving a convex optimization problem, specifically nuclear-norm minimization. They also demonstrate that matrix completion is robust to noise, with the ability to recover a low-rank matrix from a small number of noisy samples with an error proportional to the noise level. The paper discusses the connection between matrix completion and compressed sensing, highlighting that both fields rely on the assumption that the underlying signal or matrix has a certain structure (sparsity in compressed sensing, low rank in matrix completion). It also introduces the concept of incoherence, which is crucial for ensuring that the singular vectors of the matrix are spread out across all coordinates, making the matrix recoverable from a small number of samples. The authors analyze the exact recovery of low-rank matrices under noiseless conditions and then extend their results to the noisy case. They show that nuclear-norm minimization can accurately recover low-rank matrices even when the observed entries are corrupted with noise. They also provide theoretical guarantees for the accuracy of the recovery, showing that the error is proportional to the noise level. The paper includes numerical experiments that demonstrate the effectiveness of nuclear-norm minimization in recovering low-rank matrices from a small number of noisy samples. These experiments show that the method performs well in practice, even when the number of observed entries is small. The authors also compare their results with those of an oracle, which has perfect knowledge of the matrix structure. They show that the performance of nuclear-norm minimization is close to the theoretical limit achievable by an oracle, even when the matrix is only approximately low rank. In conclusion, the paper provides a detailed analysis of matrix completion, showing that it is a powerful tool for recovering low-rank matrices from a small number of entries, even in the presence of noise. The results are supported by both theoretical analysis and numerical experiments, demonstrating the practical effectiveness of the method.This paper presents a comprehensive survey of matrix completion, a problem of recovering a low-rank matrix from a small number of its entries, which may be corrupted by noise. The authors show that under suitable conditions, one can recover a low-rank matrix from a nearly minimal set of entries by solving a convex optimization problem, specifically nuclear-norm minimization. They also demonstrate that matrix completion is robust to noise, with the ability to recover a low-rank matrix from a small number of noisy samples with an error proportional to the noise level. The paper discusses the connection between matrix completion and compressed sensing, highlighting that both fields rely on the assumption that the underlying signal or matrix has a certain structure (sparsity in compressed sensing, low rank in matrix completion). It also introduces the concept of incoherence, which is crucial for ensuring that the singular vectors of the matrix are spread out across all coordinates, making the matrix recoverable from a small number of samples. The authors analyze the exact recovery of low-rank matrices under noiseless conditions and then extend their results to the noisy case. They show that nuclear-norm minimization can accurately recover low-rank matrices even when the observed entries are corrupted with noise. They also provide theoretical guarantees for the accuracy of the recovery, showing that the error is proportional to the noise level. The paper includes numerical experiments that demonstrate the effectiveness of nuclear-norm minimization in recovering low-rank matrices from a small number of noisy samples. These experiments show that the method performs well in practice, even when the number of observed entries is small. The authors also compare their results with those of an oracle, which has perfect knowledge of the matrix structure. They show that the performance of nuclear-norm minimization is close to the theoretical limit achievable by an oracle, even when the matrix is only approximately low rank. In conclusion, the paper provides a detailed analysis of matrix completion, showing that it is a powerful tool for recovering low-rank matrices from a small number of entries, even in the presence of noise. The results are supported by both theoretical analysis and numerical experiments, demonstrating the practical effectiveness of the method.