A cut-and-project perspective for linearized Bregman iterations

Abstract The linearized Bregman iterations (LBreI) method and its variants are powerful tools for finding sparse or low-rank solutions to underdetermined linear systems. In this study we propose a cut-and-project perspective for the linearized Bregman method via a bilevel optimization formulation, along with a new unified algorithmic framework. The new perspective, not only encompasses various existing LBreI variants as specific instances, but also allows us to extend the linearized Bregman method to solve more general inverse problems. We provide a completed convergence result of the proposed algorithmic framework, including convergence guarantees to feasible points and optimal solutions, and the sublinear convergence rate. Furthermore, we provide a convergence guarantee to a feasible point with a large range of step-sizes, which are wider compared with existing literature. In particular, we provide a detailed analysis of the impact of step-sizes on the algorithm and identify the precise range of step-sizes, which can guide the selection of practical values in applications. Moreover, we introduce the Bregman distance growth condition to ensure linear convergence. At last, our findings are illustrated via numerical tests.