Effective Condition Number Bounds for Convex Regularization

TL;DR

This paper establishes bounds on the condition number for convex regularization problems, linking geometric properties to statistical performance, and provides new insights into phase transitions in undersampling scenarios.

Contribution

It introduces novel bounds relating Renegar's condition number to statistical performance metrics, applicable even when analysis operators are ill-conditioned.

Findings

Bounds depend only on a projected analysis operator.

Effective bounds are obtained despite ill-conditioned operators.

New phase transition bounds for undersampling in convex regularization.

Abstract

We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell_1$-analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.