Novel min-max reformulations of Linear Inverse Problems

TL;DR

This paper introduces a new convex-concave min-max reformulation of linear inverse problems, enabling simpler algorithms for solutions and advancing methods for dictionary learning with recovery guarantees.

Contribution

It presents a novel convex-concave min-max reformulation of linear inverse problems, linking saddle points to solutions and facilitating new solution algorithms.

Findings

Characterized saddle points in terms of LIP solutions

Developed simple saddle point algorithms for LIP

Enabled new approaches to dictionary learning with recovery constraints

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.