# Harmonic Mean Iteratively Reweighted Least Squares for Low-Rank Matrix Recovery

### Abstract

We propose a new iteratively reweighted least squares (IRLS) algorithm for the recovery of a matrix $X \in \mathbb{C}^{d_1 \times d_2}$ of rank $r \ll \min(d_1,d_2)$ from incomplete linear observations, solv- ing a sequence of low complexity linear problems. The easily implementable algorithm, which we call harmonic mean iteratively reweighted least squares $\texttt{HM-IRLS}$, optimizes a non-convex Schatten-$p$ quasi-norm penalization to promote low-rankness and carries three major strengths, in particular for the matrix completion setting. First, we observe a remarkable global convergence behavior of the algorithm’s iterates to the low-rank matrix for relevant, interesting cases, for which any other state-of-the-art optimization approach fails the recovery. Secondly, $\texttt{HM-IRLS}$ exhibits an empirical recovery probability close to $1$ even for a number of measurements very close to the theoretical lower bound $r(d_1 +d_2+r)$, i.e., already for significantly fewer linear observations than any other tractable approach in the literature. Thirdly, $\texttt{HM-IRLS}$ exhibits a locally superlinear rate of convergence (of order $2-p$) if the linear observations fulfill a suitable null space property. While for the first two properties we have so far only strong empirical evidence, we prove the third property as our main theoretical result.

Type
Publication
Journal of Machine Learning Research (JMLR), 19(47), 1815-1863

A preliminary version of this work was presented at SampTA 2017, see here for the conference paper.