Universal Matrix Completion

TL;DR

This paper introduces a universal matrix completion method that guarantees exact recovery under diverse sampling schemes, including non-uniform and structured sampling, with improved sample complexity.

Contribution

It provides the first universal recovery guarantee for matrix completion applicable to various sampling schemes, including graph-based sampling, with reduced sample complexity.

Findings

Exact recovery for matrices with diverse sampling schemes

Recovery guaranteed under spectral gap conditions of sampling graphs

Reduced sample complexity of $O(nr^2)$ for certain incoherent matrices

Abstract

The problem of low-rank matrix completion has recently generated a lot of interest leading to several results that offer exact solutions to the problem. However, in order to do so, these methods make assumptions that can be quite restrictive in practice. More specifically, the methods assume that: a) the observed indices are sampled uniformly at random, and b) for every new matrix, the observed indices are sampled afresh. In this work, we address these issues by providing a universal recovery guarantee for matrix completion that works for a variety of sampling schemes. In particular, we show that if the set of sampled indices come from the edges of a bipartite graph with large spectral gap (i.e. gap between the first and the second singular value), then the nuclear norm minimization based method exactly recovers all low-rank matrices that satisfy certain incoherence properties. Moreover, we also show that under certain stricter incoherence conditions, $O(nr^2)$ uniformly sampled entries are enough to recover any rank-$r$ $n\times n$ matrix, in contrast to the $O(nr\log n)$ sample complexity required by other matrix completion algorithms as well as existing analyses of the nuclear norm method.