Revisiting Linearized Bregman Iterations under Lipschitz-like Convexity Condition

TL;DR

This paper extends the applicability of linearized Bregman iterations by replacing the gradient Lipschitz continuity assumption with a more general Lipschitz-like convexity condition, enabling broader use in convex and nonconvex inverse problems.

Contribution

It introduces a unified framework that replaces the Lipschitz continuity assumption with a Lipschitz-like convexity condition for LBreI-type methods, applicable to both convex and nonconvex problems.

Findings

The Lipschitz-like convexity condition generalizes the gradient Lipschitz continuity.

The framework applies effectively to linear and quadratic inverse problems.

Theoretical analysis supports broader applicability of LBreI methods.

Abstract

The linearized Bregman iterations (LBreI) and its variants have received considerable attention in signal/image processing and compressed sensing. Recently, LBreI has been extended to a larger class of nonconvex functions, along with several theoretical issues left for further investigation. In particular, the gradient Lipschitz continuity assumption precludes its use in many practical applications. In this study, we propose a generalized algorithmic framework to unify LBreI-type methods. Our main discovery is that the gradient Lipschitz continuity assumption can be replaced by a Lipschitz-like convexity condition in both convex and nonconvex cases. The proposed framework and theory are then applied to linear/quadratic inverse problems.