Nonlinear Residual Minimization by Iteratively Reweighted Least Squares

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.Comment: 37 pages. arXiv admin note: text overlap with arXiv:0807.0575 by other author

Inexact Alternating Optimization for Phase Retrieval In the Presence of Outliers

Phase retrieval has been mainly considered in the presence of Gaussian noise. However, the performance of the algorithms proposed under the Gaussian noise model severely degrades when grossly corrupted data, i.e., outliers, exist. This paper investigates techniques for phase retrieval in the presence of heavy-tailed noise -- which is considered a better model for situations where outliers exist. An $\ell_p$-norm ($0<p<2$) based estimator is proposed for fending against such noise, and two-block inexact alternating optimization is proposed as the algorithmic framework to tackle the resulting optimization problem. Two specific algorithms are devised by exploring different local approximations within this framework. Interestingly, the core conditional minimization steps can be interpreted as iteratively reweighted least squares and gradient descent. Convergence properties of the algorithms are discussed, and the Cram\'er-Rao bound (CRB) is derived. Simulations demonstrate that the proposed algorithms approach the CRB and outperform state-of-the-art algorithms in heavy-tailed noise.Comment: 23 pages, 16 figure