Exact solutions of infinite dimensional total-variation regularized problems

TL;DR

This paper characterizes the structure and provides a method for computing exact solutions to infinite dimensional total-variation regularized inverse problems over Banach spaces, extending current understanding in the field.

Contribution

It demonstrates the existence of sparse solutions and introduces a way to compute exact solutions via finite dimensional convex programs.

Findings

Existence of m-sparse solutions under certain conditions

Exact solutions can be obtained without discretization

Extends recent advances in total-variation regularized problems

Abstract

We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary convex function. The first contribution is about the solu-tion's structure: we show that under suitable assumptions, there always exist an m-sparse solution, where m is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exacts solutions of the infinite dimensional problem can be obtained by solving two consecutive finite dimensional convex programs. These results extend recent advances in the understanding of total-variation reg-ularized problems.