Maximal Solutions of Sparse Analysis Regularization

TL;DR

This paper investigates the structure of solutions to analysis-Lasso regularization, revealing that maximal D-support solutions are interior points of the solution set and proposing an algorithm to find them.

Contribution

It offers a geometric interpretation of maximal solutions and introduces a primal-dual interior point method to identify such solutions.

Findings

Maximal solutions reside in the relative interior of the solution set.

A primal-dual interior point algorithm can exhibit maximal solutions.

Provides geometric insight into the non-uniqueness of analysis-Lasso solutions.

Abstract

This paper deals with the non-uniqueness of the solutions of an analysis-Lasso regularization. Most of previous works in this area is concerned with the case where the solution set is a singleton, or to derive guarantees to enforce uniqueness. Our main contribution consists in providing a geometrical interpretation of a solution with a maximal D-support, namely the fact that such a solution lives in the relative interior of the solution set. With this result in hand, we also provide a way to exhibit a maximal solution using a primal-dual interior point algorithm.