Low-Rank Matrix Completion

While datasets are frequently represented as matrices, real-word data is imperfect and entries are often missing. In many cases, the data are very sparse and the matrix must be filled in before any subsequent work can be done. This optimization problem, known as matrix completion, can be made well-defined by assuming the matrix to be low rank. The resulting rank-minimization problem is NP-hard, but it has recently been shown that the rank constraint can be replaced with a nuclear norm constraint and, with high probability, the global minimum of the problem will not change. Because this nuclear norm problem is convex and can be optimized efficiently, there has been a significant amount of research over the past few years to develop optimization algorithms that perform well. In this report, we review several methods for low-rank matrix completion. The first paper we review presents an iterative algorithm to efficiently complete extremely large matrices. The second paper formulates the problem directly as matrix factorization, which can be optimized using several different methods. Next, we present an algorithm that changes the way that the optimization is carried out in order to achieve better convergence on ill-conditioned matrices. Finally, we describe a related online algorithm for matrix completion. We show how these algorithms are related and directly compare them using experiments on synthetic data.