Matrix estimation by Universal Singular Value Thresholding

TL;DR

This paper introduces USVT, a simple yet effective matrix estimation method that achieves near-optimal error rates across various structured matrix problems, including low rank, blockmodels, and graphon estimation.

Contribution

The paper proposes USVT, a universal singular value thresholding technique that attains minimax error rates for a broad class of structured matrix estimation problems.

Findings

USVT achieves minimax error rate up to a constant factor.

Applicable to low rank, blockmodels, and graphon estimation.

Performs well across diverse matrix estimation tasks.

Abstract

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.