The Linearized Bregman Method via Split Feasibility Problems: Analysis and Generalizations

TL;DR

This paper reformulates the linearized Bregman method as a split feasibility problem, providing a unified convergence analysis and extending its applicability to various objective functions, noise models, and constraints.

Contribution

It introduces a new algorithmic framework based on Bregman projections for the linearized Bregman method and proves its convergence, enabling several generalizations.

Findings

Convergence of the proposed framework is established.

The approach generalizes the linearized Bregman method to other objectives.

Extensions include handling non-Gaussian noise and box constraints.

Abstract

The linearized Bregman method is a method to calculate sparse solutions to systems of linear equations. We formulate this problem as a split feasibility problem, propose an algorithmic framework based on Bregman projections and prove a general convergence result for this framework. Convergence of the linearized Bregman method will be obtained as a special case. Our approach also allows for several generalizations such as other objective functions, incremental iterations, incorporation of non-gaussian noise models or box constraints.