The Geometry and Well-Posedness of Sparse Regularized Linear Regression

TL;DR

This paper investigates the geometric conditions ensuring the existence, uniqueness, and stability of solutions in sparse regularized linear regression problems with convex piecewise linear regularizers, highlighting their computational complexity.

Contribution

It introduces a geometric framework for analyzing well-posedness in sparse regularized regression with polyhedral regularizers and compares these conditions to smooth regularization.

Findings

Geometric conditions for well-posedness are established.

Comparison between polyhedral and smooth regularization conditions.

Verification of conditions is computationally challenging.

Abstract

In this work, we study the well-posedness of certain sparse regularized linear regression problems, i.e., the existence, uniqueness and continuity of the solution map with respect to the data. We focus on regularization functions that are convex piecewise linear, i.e., whose epigraph is polyhedral. This includes total variation on graphs and polyhedral constraints. We provide a geometric framework for these functions based on their connection to polyhedral sets and apply this to the study of the well-posedness of the corresponding sparse regularized linear regression problem. Particularly, we provide geometric conditions for well-posedness of the regression problem, compare these conditions to those for smooth regularization, and show the computational difficulty of verifying these conditions.