Convergence analysis of the FOCUSS algorithm

TL;DR

This paper provides a comprehensive convergence analysis of the FOCUSS algorithm, including derivation, proof of convergence, and rate analysis based on the sparsity parameter, supported by numerical experiments.

Contribution

It offers the first systematic convergence theory for FOCUSS, including derivation, convergence proof, and rate analysis with numerical validation.

Findings

Proved convergence of FOCUSS algorithm.

Analyzed convergence rate relative to sparsity parameter p.

Validated convergence rate through numerical experiments.

Abstract

FOCal Underdetermined System Solver (FOCUSS) is a powerful tool for sparse representation and underdetermined inverse problems, which is extremely easy to implement. In this paper, we give a comprehensive convergence analysis on the FOCUSS algorithm towards establishing a systematic convergence theory by providing three primary contributions as follows. First, we give a rigorous derivation for this algorithm exploiting the auxiliary function. Then, we prove its convergence. Third, we systematically study its convergence rate with respect to the sparsity parameter p and demonstrate its convergence rate by numerical experiments.