# Convex Geometry of the Generalized Matrix-Fractional Function

## TL;DR

This paper simplifies the geometric and subdifferential representations of generalized matrix-fractional functions, enabling easier computation of related geometric objects crucial for optimization and inverse problems.

## Contribution

It introduces simplified support function and subdifferential representations for GMF functions, facilitating advanced geometric analysis.

## Key findings

- Simplified support function representation for GMF functions
- Simplified subdifferential representation for GMF functions
- Enables computation of new geometric objects related to GMF functions

## Abstract

Generalized matrix-fractional (GMF) functions are a class of matrix support functions introduced by Burke and Hoheisel as a tool for unifying a range of seemingly divergent matrix optimization problems associated with inverse problems, regularization and learning. In this paper we dramatically simplify the support function representation for GMF functions as well as the representation of their subdifferentials. These new representations allow the ready computation of a range of important related geometric objects whose formulations were previously unavailable.