Inexact FPPA for the $\ell_0$ Sparse Regularization Problem

TL;DR

This paper develops inexact fixed-point proximity algorithms for non-convex $\, ext{ extlbrackdbl}\,0 ext{ extgreater brackdbl}$ sparse regularization, ensuring convergence to local minimizers even when exact proximity operators are unavailable.

Contribution

It introduces a convergence-guaranteed inexact algorithm for $\, ext{ extlbrackdbl}\,0 extgreater brackdbl$ models, applicable when closed-form proximity operators are not accessible.

Findings

The algorithm converges to local minimizers of the $\, ext{ extlbrackdbl}\,0 extgreater brackdbl$ problem.

Local minimizers found outperform $\, ext{ extlbrackdbl}\,1 extgreater brackdbl$ models in accuracy and sparsity.

Numerical experiments in machine learning and image processing validate the approach.

Abstract

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the $\ell_0$ norm. Specifically, the $\ell_0$ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the $\ell_0$ norm regularization term. Such an $\ell_0$ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the resulting inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the $\ell_0$ models found by the proposed inexact algorithm outperform global minimizers of the corresponding $\ell_1$ models, in terms of approximation accuracy and sparsity of the solutions.