On Convex Duality in Linear Inverse Problems

TL;DR

This paper explores convex duality in ill-posed linear inverse problems, introducing a novel min-max reformulation that enables simple algorithms and advances dictionary learning with recovery guarantees.

Contribution

It presents a new convex-concave min-max reformulation of linear inverse problems, facilitating algorithm development and applications to dictionary learning with recovery constraints.

Findings

Introduces a convex-concave min-max reformulation.

Develops simple ascend-descent algorithms for LIP.

Enables dictionary learning with almost sure recovery.

Abstract

In this article we dwell into the class of so called ill posed Linear Inverse Problems (LIP) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from its relatively few random linear measurements. Such problems arise in variety of settings with applications ranging from medical image processing, recommender systems etc. We provide an exposition to the convex duality of the linear inverse problems, and obtain a novel and equivalent convex-concave min-max reformulation that gives rise to simple ascend-descent type algorithms to solve an LIP. Moreover, such a reformulation is crucial in developing methods to solve the dictionary learning problem with almost sure recovery constraints.