Matrix estimation by Universal Singular Value Thresholding

Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Cand\`es and collaborators. Typically, it is assumed that the underlying matrix has low rank. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has `a little bit of structure'. In particular, the matrix need not be of low rank. The procedure is very simple and fast, works under minimal assumptions, and is applicable for very large matrices. Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to give simple solutions to difficult questions in low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, problems related to graph limits, and generalized Bradley-Terry models for pairwise comparison.