Robust Polyhedral Regularization

TL;DR

This paper demonstrates that polyhedral regularization methods for linear inverse problems are robust to noise, providing conditions under which the true solution's geometric face is correctly recovered, with error proportional to noise level.

Contribution

It introduces a general robustness criterion for polyhedral regularization, extending known results for $ ext{l}^1$ regularization to broader regularizations including $ ext{l}^ extinfty$ and $ ext{l}^1$-$ extinfty$.

Findings

Recovery error is proportional to noise level.

The criterion applies to various polyhedral regularizations.

Guarantees extend to $ ext{l}^ extinfty$ and $ ext{l}^1$-$ extinfty$ regularizations.

Abstract

In this paper, we establish robustness to noise perturbations of polyhedral regularization of linear inverse problems. We provide a sufficient condition that ensures that the polyhedral face associated to the true vector is equal to that of the recovered one. This criterion also implies that the $\ell^2$ recovery error is proportional to the noise level for a range of parameter. Our criterion is expressed in terms of the hyperplanes supporting the faces of the unit polyhedral ball of the regularization. This generalizes to an arbitrary polyhedral regularization results that are known to hold for sparse synthesis and analysis $\ell^1$ regularization which are encompassed in this framework. As a byproduct, we obtain recovery guarantees for $\ell^\infty$ and $\ell^1-\ell^\infty$ regularization.