# A Non-Convex Relaxation for Fixed-Rank Approximation

## TL;DR

This paper introduces a non-convex relaxation method for fixed-rank matrix approximation that avoids the bias of nuclear norm approaches and often converges to better solutions, especially under RIP conditions.

## Contribution

It proposes a novel non-convex relaxation technique for low-rank matrix approximation that reduces bias and demonstrates favorable convergence properties compared to nuclear norm methods.

## Key findings

- The non-convex relaxation often has a single local minimizer under RIP.
- Numerical tests show better solutions than nuclear norm methods.
- The approach performs well even when RIP does not hold.

## 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.