Structured Sparsity Promoting Functions

TL;DR

This paper introduces a new class of structured semiconvex sparsity promoting functions derived from convex functions and their Moreau envelopes, with theoretical guarantees and analysis of their proximity operators.

Contribution

It presents a novel scheme to construct structured sparsity functions with proven properties and explores their proximity operators, extending existing methods in high-dimensional variable selection.

Findings

Sparsity guarantees for the new family of functions

Analysis of proximity operators for various special functions

Examples demonstrating the applicability of the proposed functions

Abstract

Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we provide sparsity guarantees for the general family of functions. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and the linear combinations of them. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases.