Efficient Computation of the Non-convex Quasi-norm Ball Projection with Iterative Reweighted Approach

TL;DR

This paper introduces a new localized approximation method for efficiently projecting onto the non-convex $\,\ell_p$ quasi-norm ball, improving convergence and computational efficiency over existing algorithms.

Contribution

It proposes a novel approximation technique and enhances the iterative reweighted algorithm for better accuracy and efficiency in $\,\ell_p$ quasi-norm ball projection.

Findings

The method converges globally.

It outperforms previous algorithms in speed.

Numerical results confirm improved accuracy.

Abstract

In this study, we focus on computing the projection onto the $\ell_p$ quasi-norm ball, which is challenging due to the non-convex and non-Lipschitz nature inherent in the $\ell_p$ quasi-norm with $0<p<1$. We propose a novel localized approximation method that yields a Lipschitz continuous concave surrogate function for the $\ell_p$ quasi-norm with improved approximation quality. Building on this approximation, we enhance the state-of-the-art iterative reweighted algorithm proposed by Yang et al. (J Mach Learn Res 23:1-31, 2022) by constructing tighter subproblems. This improved algorithm solves the $\ell_p$ quasinorm ball projection problem through a series of tractable projections onto the weighted $\ell_1$ norm balls. Convergence analyses and numerical studies demonstrate the global convergence and superior computational efficiency of the proposed method.