Block-wise Minimization-Majorization algorithm for Huber's criterion: sparse learning and applications

TL;DR

This paper introduces a block-wise minimization-majorization algorithm for Huber's criterion, enhancing robustness and convergence in joint estimation, and applies it to sparse learning and image denoising.

Contribution

It reformulates Huber's original algorithm within a block-wise minimization framework and proposes adaptive step sizes, improving robustness and convergence in sparse learning.

Findings

Improved convergence with data-adaptive step sizes.

Effective sparse learning in underdetermined linear models.

Successful application to image denoising.

Abstract

Huber's criterion can be used for robust joint estimation of regression and scale parameters in the linear model. Huber's (Huber, 1981) motivation for introducing the criterion stemmed from non-convexity of the joint maximum likelihood objective function as well as non-robustness (unbounded influence function) of the associated ML-estimate of scale. In this paper, we illustrate how the original algorithm proposed by Huber can be set within the block-wise minimization majorization framework. In addition, we propose novel data-adaptive step sizes for both the location and scale, which are further improving the convergence. We then illustrate how Huber's criterion can be used for sparse learning of underdetermined linear model using the iterative hard thresholding approach. We illustrate the usefulness of the algorithms in an image denoising application and simulation studies.