Perspective Functions: Proximal Calculus and Applications in High-Dimensional Statistics

TL;DR

This paper introduces a systematic approach using proximal methods to efficiently solve optimization problems involving perspective functions, which are common in high-dimensional statistical analysis.

Contribution

It develops a general framework for constructing proximity operators of perspective functions and applies these to create new algorithms for high-dimensional regression and variable selection.

Findings

Proximity operators for perspective functions can be explicitly computed or approximated efficiently.

New proximal algorithms improve high-dimensional regression and variable selection tasks.

Framework unifies and extends existing methods in nonsmooth optimization.

Abstract

Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems. In this paper, we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions. We study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations. These results constitute central building blocks in the design of proximal optimization algorithms. We showcase the versatility of the framework by designing novel proximal algorithms for state-of-the-art regression and variable selection schemes in high-dimensional statistics.