Model Consistency of Partly Smooth Regularizers

TL;DR

This paper investigates the stability and consistency of solutions obtained from least-square regression with partly smooth convex regularizers, demonstrating conditions under which the model correctly identifies the underlying low-dimensional structure despite noise.

Contribution

It introduces a generalized irrepresentable condition ensuring stable model selection and consistency for a broad class of partly smooth regularizers, unifying previous results.

Findings

Generalized irrepresentable condition guarantees model stability.

Model consistency is achieved as the number of measurements grows.

Condition is nearly necessary for stable model selection.

Abstract

This paper studies least-square regression penalized with partly smooth convex regularizers. This class of functions is very large and versatile allowing to promote solutions conforming to some notion of low-complexity. Indeed, they force solutions of variational problems to belong to a low-dimensional manifold (the so-called model) which is stable under small perturbations of the function. This property is crucial to make the underlying low-complexity model robust to small noise. We show that a generalized "irrepresentable condition" implies stable model selection under small noise perturbations in the observations and the design matrix, when the regularization parameter is tuned proportionally to the noise level. This condition is shown to be almost a necessary condition. We then show that this condition implies model consistency of the regularized estimator. That is, with a probability tending to one as the number of measurements increases, the regularized estimator belongs to the correct low-dimensional model manifold. This work unifies and generalizes several previous ones, where model consistency is known to hold for sparse, group sparse, total variation and low-rank regularizations.