Variational Properties of Matrix Functions via the Generalized Matrix-Fractional Function

TL;DR

This paper introduces a unified framework using the generalized matrix fractional function to represent and analyze convex matrix functions, enabling new computational and optimization techniques.

Contribution

It provides a novel representation of convex matrix functions via the GMF, facilitating analysis, computation, and smoothing approaches for a broad class of functions.

Findings

Representation of convex matrix functions as partial infimal projections

Formulas for conjugates and subdifferentials of these functions

Foundation for new smoothing optimization methods

Abstract

We show that many important convex matrix functions can be represented as the partial infimal projection of the generalized matrix fractional (GMF) and a relatively simple convex function. This representation provides conditions under which such functions are closed and proper as well as formulas for the ready computation of both their conjugates and subdifferentials. Special attention is given to support and indicator functions. Particular instances yield all weighted Ky Fan norms and squared gauges on $\mathbb R^{n\times m}$, and as an example we show that all variational Gram functions are representable as squares of gauges. Other instances yield weighted sums of the Frobenius and nuclear norms. The scope of applications is large and the range of variational properties and insight is fascinating and fundamental. An important byproduct of these representations is that they lay the foundation for a smoothing approach to many matrix functions on the interior of the domain of the GMF function, which opens the door to a range of unexplored optimization methods.