Juliane Sigl

Christian Kümmerle, Juliane Sigl

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, solving a sequence of low complexity linear problems. The easily implementable algorithm, which we call harmonic mean iteratively reweighted least squares (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, 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, 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.

Juliane Sigl

We address the numerical solution of minimal norm residuals of {\it nonlinear} equations in finite dimensions. We take inspiration from the problem of finding a sparse vector solution by using greedy algorithms based on iterative residual minimizations in the $\ell_p$-norm, for $1 \leq p \leq 2$. Due to the mild smoothness of the problem, especially for $p \to 1$, we develop and analyze a generalized version of Iteratively Reweighted Least Squares (IRLS). This simple and efficient algorithm performs the solution of optimization problems involving non-quadratic possibly non-convex and non-smooth cost functions, which can be transformed into a sequence of common least squares problems, which can be tackled more efficiently.While its analysis has been developed in many contexts when the model equation is {\it linear}, no results are provided in the {\it nonlinear} case. We address the convergence and the rate of error decay of IRLS for nonlinear problems. The convergence analysis is based on its reformulation as an alternating minimization of an energy functional, whose variables are the competitors to solutions of the intermediate reweighted least squares problems. Under specific conditions of coercivity and local convexity, we are able to show convergence of IRLS to minimizers of the nonlinear residual problem. For the case where we are lacking local convexity, we propose an appropriate convexification.. To illustrate the theoretical results we conclude the paper with several numerical experiments. We compare IRLS with standard Matlab functions for an easily presentable example and numerically validate our theoretical results in the more complicated framework of phase retrieval problems. Finally we examine the recovery capability of the algorithm in the context of data corrupted by impulsive noise where the sparsification of the residual is desired.