On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators

TL;DR

This paper explores the geometric structure of spaces of group equivariant non-expansive operators, showing they can be modeled as Riemannian manifolds to facilitate optimization and selection of representative operators.

Contribution

It introduces a Riemannian manifold structure on the space of group equivariant non-expansive operators, enabling gradient-based optimization and operator selection.

Findings

The space of operators can be endowed with a Riemannian structure.

Gradient descent methods can be applied on this manifold.

A procedure for selecting representative operators is proposed.

Abstract

Group equivariant non-expansive operators have been recently proposed as basic components in topological data analysis and deep learning. In this paper we study some geometric properties of the spaces of group equivariant operators and show how a space $\mathcal{F}$ of group equivariant non-expansive operators can be endowed with the structure of a Riemannian manifold, so making available the use of gradient descent methods for the minimization of cost functions on $\mathcal{F}$. As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant non-expansive operators in the considered manifold.