Exact Matrix Completion via Convex Optimization

TL;DR

This paper proves that most low-rank matrices can be exactly recovered from a small number of randomly sampled entries using convex optimization, extending ideas from compressed sensing to matrix completion.

Contribution

It establishes theoretical guarantees for exact matrix recovery via nuclear norm minimization with near-optimal sampling conditions.

Findings

Most low-rank matrices can be recovered with high probability.

Recovery is guaranteed if the number of samples exceeds a threshold proportional to n^{1.2} r log n.

The results extend to all ranks with a slightly higher sample complexity.

Abstract

We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys m >= C n^{1.2} r log n for some positive numerical constant C, then with very high probability, most n by n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.