Orbital Geometry in Optimisation

TL;DR

This paper explores how group symmetries can be leveraged in optimisation to generalise classic subdifferential characterisations and simplify projections onto symmetric sets, with applications in sparse signal recovery.

Contribution

It generalises Lewis's subdifferential result to Schur convex functions invariant under reflection groups and extends projection results to sparsity constraints.

Findings

Generalisation of subdifferential characterisation for Schur convex functions

Simplified projection methods for symmetric sets

Applications to sparse signal recovery and compressed sensing

Abstract

We discuss the use of group symmetries in optimisation, in particular with respect to the structure of subdifferential and projection operators. This allows us to generalise a classic result of Adrian Lewis regarding the characterisation of the subdifferential of a permutation invariant convex function to the characterisation of the proximal subdifferential of a Schur convex function that is invariant with respect to a finite reflection group. We are also able to simplify and generalise results on projections onto symmetric sets, in particular, we study projections on sparsity constraints used in sparse signal recovery and compressed sensing.