Novel min-max reformulations of Linear Inverse Problems
Abstract
In this article, we dwell into the class of so-called ill-posed Linear Inverse Problems (LIP) which simply refers to the task of recovering the entire signal from its relatively few random linear measurements. Such problems arise in a variety of settings with applications ranging from medical image processing, recommender systems, etc. We propose a slightly generalized version of the error constrained linear inverse problem and obtain a novel and equivalent convex-concave min-max reformulation by providing an exposition to its convex geometry. Saddle points of the min-max problem are completely characterized in terms of a solution to the LIP, and vice versa. Applying simple saddle point seeking ascend-descent type algorithms to solve the min-max problems provides novel and simple algorithms to find a solution to the LIP. Moreover, the reformulation of an LIP as the min-max problem provided in this article is crucial in developing methods to solve the dictionary learning problem with almost sure recovery constraints.
Cite
@article{arxiv.2007.02448,
title = {Novel min-max reformulations of Linear Inverse Problems},
author = {Mohammed Rayyan Sheriff and Debasish Chatterjee},
journal= {arXiv preprint arXiv:2007.02448},
year = {2022}
}