A quasi-Newton proximal splitting method

TL;DR

This paper introduces a quasi-Newton proximal splitting method that accelerates convex minimization problems by leveraging new convex analysis results, efficient proximity operator computations, and dual problem structures, with broad applications in signal processing and machine learning.

Contribution

It presents a novel quasi-Newton method for convex optimization that improves efficiency by exploiting scaled norms and dual problem piece-wise linearity.

Findings

The method accelerates convergence compared to existing algorithms.

Efficient proximity operator implementations are developed for specific function classes.

The approach has broad applications in signal processing, sparse recovery, and machine learning.

Abstract

A new result in convex analysis on the calculation of proximity operators in certain scaled norms is derived. We describe efficient implementations of the proximity calculation for a useful class of functions; the implementations exploit the piece-wise linear nature of the dual problem. The second part of the paper applies the previous result to acceleration of convex minimization problems, and leads to an elegant quasi-Newton method. The optimization method compares favorably against state-of-the-art alternatives. The algorithm has extensive applications including signal processing, sparse recovery and machine learning and classification.