# The Geometry of Sparse Analysis Regularization

## TL;DR

This paper explores the geometric structure of solution sets in analysis l1-regularization problems, revealing how support, sign patterns, and sub-polyhedra relate to the solution space without assuming uniqueness.

## Contribution

It provides a detailed geometric analysis of the solution set, including support and sign pattern characterizations, and shows how sub-polyhedra can be realized as solution sets with explicit parameters.

## Key findings

- Support of solutions linked to minimal faces of the polyhedron
- Extremal points can be identified via an algebraic test
- Any sub-polyhedron of the level set can be realized as a solution set

## Abstract

Analysis sparsity is a common prior in inverse problem or machine learning including special cases such as Total Variation regularization, Edge Lasso and Fused Lasso. We study the geometry of the solution set (a polyhedron) of the analysis l1-regularization (with l2 data fidelity term) when it is not reduced to a singleton without any assumption of the analysis dictionary nor the degradation operator. In contrast with most theoretical work, we do not focus on giving uniqueness and/or stability results, but rather describe a worst-case scenario where the solution set can be big in terms of dimension. Leveraging a fine analysis of the sub-level set of the regularizer itself, we draw a connection between support of a solution and the minimal face containing it, and in particular prove that extremal points can be recovered thanks to an algebraic test. Moreover, we draw a connection between the sign pattern of a solution and the ambient dimension of the smallest face containing it. Finally, we show that any arbitrary sub-polyhedra of the level set can be seen as a solution set of sparse analysis regularization with explicit parameters.