A Non-Convex Relaxation for Fixed-Rank Approximation
Abstract
This paper considers the problem of finding a low rank matrix from observations of linear combinations of its elements. It is well known that if the problem fulfills a restricted isometry property (RIP), convex relaxations using the nuclear norm typically work well and come with theoretical performance guarantees. On the other hand these formulations suffer from a shrinking bias that can severely degrade the solution in the presence of noise. In this theoretical paper we study an alternative non-convex relaxation that in contrast to the nuclear norm does not penalize the leading singular values and thereby avoids this bias. We show that despite its non-convexity the proposed formulation will in many cases have a single local minimizer if a RIP holds. Our numerical tests show that our approach typically converges to a better solution than nuclear norm based alternatives even in cases when the RIP does not hold.
Cite
@article{arxiv.1706.05855,
title = {A Non-Convex Relaxation for Fixed-Rank Approximation},
author = {Carl Olsson and Marcus Carlsson and Erik Bylow},
journal= {arXiv preprint arXiv:1706.05855},
year = {2017}
}