A projection proximal-point algorithm for l^1-minimization

TL;DR

This paper introduces a new projection proximal-point algorithm for -minimization in infinite-dimensional Hilbert spaces, demonstrating its simplicity and potential advantages over existing methods.

Contribution

The paper proposes a novel -minimization algorithm combining proximal-point and projection steps, with convergence analysis in infinite-dimensional settings.

Findings

Algorithm performs well in experiments compared to existing methods.

The method is simple and easy to implement.

Projection proximal-point approach shows promise for -minimization.

Abstract

The problem of the minimization of least squares functionals with $\ell^1$ penalties is considered in an infinite dimensional Hilbert space setting. While there are several algorithms available in the finite dimensional setting there are only a few of them which come with a proper convergence analysis in the infinite dimensional setting. In this work we provide an algorithm from a class which have not been considered for $\ell^1$ minimization before, namely a proximal-point method in combination with a projection step. We show that this idea gives a simple and easy to implement algorithm. We present experiments which indicate that the algorithm may perform better than other algorithms if we employ them without any special tricks. Hence, we may conclude that the projection proximal-point idea is a promising idea in the context of $\ell^1$-minimization.