Convex Analysis in Spectral Decomposition Systems

Abstract

This work is concerned with the convex analysis of functions defined on (not necessarily finite-dimensional) Hilbert spaces whose values depend solely on a certain ``spectrum'' of the arguments, a class we term ``spectral functions.'' We propose a notion of a spectral decomposition system which brings together a wide array of settings underlying important applications such as Fourier-phase-invariant functions, mixed-norm regularization, and functions of eigenvalues or (signed) singular values of matrices. We are particularly motivated by algorithmic requirements for evaluating convex analytical objects. Thus, a central contribution is a novel reduced minimization principle that enables the constructive reduction of minimization problems involving spectral functions to those of the simpler associated invariant functions. This result is then leveraged to explicitly evaluate the conjugates, subgradients, and set-valued Bregman proximity operators of spectral functions.