  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\`{e}s and collaborators.
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." Surprisingly, this simple estimator achieves the minimax error rate
up to a constant factor. The method is applied to solve problems related to low
rank matrix estimation, blockmodels, distance matrix completion, latent space
models, positive definite matrix completion, graphon estimation and generalized
Bradley--Terry models for pairwise comparison.
