This paper addresses the problem of recovering an unknown matrix from a small fraction of its entries, known as matrix completion. The authors present optimal results quantifying the minimum number of entries needed to recover a matrix of rank \( r \) exactly using any method (the information-theoretic limit). They show that under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convex program as soon as the number of entries is on the order of the information-theoretic limit (up to logarithmic factors). The convex program finds the matrix with the minimum nuclear norm among all matrices consistent with the observed entries. Specifically, the paper demonstrates that \( nr \log(n) \) samples are needed to recover a random \( n \times n \) matrix of rank \( r \), and nuclear norm minimization succeeds when the number of entries is of the form \( nr \text{polylog}(n) \). The authors also introduce three model matrices—uniformly bounded, low-rank low-coherence, and random orthogonal—to illustrate that many matrices of interest obey the strong incoherence property with a small parameter \( \mu \), allowing for efficient recovery. The paper concludes with a lower bound theorem, highlighting the fundamental role of coherence in controlling the information-theoretic limits of matrix completion.This paper addresses the problem of recovering an unknown matrix from a small fraction of its entries, known as matrix completion. The authors present optimal results quantifying the minimum number of entries needed to recover a matrix of rank \( r \) exactly using any method (the information-theoretic limit). They show that under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convex program as soon as the number of entries is on the order of the information-theoretic limit (up to logarithmic factors). The convex program finds the matrix with the minimum nuclear norm among all matrices consistent with the observed entries. Specifically, the paper demonstrates that \( nr \log(n) \) samples are needed to recover a random \( n \times n \) matrix of rank \( r \), and nuclear norm minimization succeeds when the number of entries is of the form \( nr \text{polylog}(n) \). The authors also introduce three model matrices—uniformly bounded, low-rank low-coherence, and random orthogonal—to illustrate that many matrices of interest obey the strong incoherence property with a small parameter \( \mu \), allowing for efficient recovery. The paper concludes with a lower bound theorem, highlighting the fundamental role of coherence in controlling the information-theoretic limits of matrix completion.