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

Pasquale Cascarano, Patrizio Frosini, Nicola Quercioli, Amir Saki

TL;DR

This work investigates the geometric and Riemannian structure of spaces of group equivariant non-expansive operators (GENEOs) and proves that, under a compact signal space equipped with a probability measure, the GENEO space is compact in the $L^2$ sense. It introduces a probability-weighted metric framework and shows how finite-dimensional GENEO submanifolds inherit a coherent Riemannian structure, enabling gradient-descent optimization on operators. The authors provide a concrete procedure to select a representative finite set of GENEOs by optimizing a nonconvex $L^2$ objective, and demonstrate this with a torus-based discretization tied to EMNIST-like signals. Overall, the paper offers a principled approach to reducing operator-space complexity for applications in topological data analysis and deep learning, with avenues for extending to broader equivariance settings and alternative signal metrics.

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.

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

TL;DR

This work investigates the geometric and Riemannian structure of spaces of group equivariant non-expansive operators (GENEOs) and proves that, under a compact signal space equipped with a probability measure, the GENEO space is compact in the sense. It introduces a probability-weighted metric framework and shows how finite-dimensional GENEO submanifolds inherit a coherent Riemannian structure, enabling gradient-descent optimization on operators. The authors provide a concrete procedure to select a representative finite set of GENEOs by optimizing a nonconvex objective, and demonstrate this with a torus-based discretization tied to EMNIST-like signals. Overall, the paper offers a principled approach to reducing operator-space complexity for applications in topological data analysis and deep learning, with avenues for extending to broader equivariance settings and alternative signal metrics.

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 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 . 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.

Paper Structure

This paper contains 10 sections, 19 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical setting
  3. Probability inner product $G$-spaces and spaces of admissible signals.
  4. The pseudo-metric on $X$ induced by admissible signals.
  5. $\Phi$-preserving homeomorphisms.
  6. Group equivariant operators.
  7. The action of $G$ on the space $\mathcal{F}^{\mathrm{all}}$ of all GENEOs.
  8. Submanifolds of GENEOs.
  9. An application of our model to select GENEOs
  10. Discussion

Key Result

Lemma 2.2

Let $x_1,x_2\in X$. Then, the map $\xi_{x_1,x_2}:\Phi\to\mathbb{R}$ defined by $\xi_{x_1,x_2}(\varphi)=|\varphi(x_1)-\varphi(x_2)|$ for any $\varphi\in\Phi$, is continuous with respect to the topology $\tau_\Phi$ on $\Phi$ induced by the topology $\tau_V$ on $V$, and it is an integrable random varia

Figures (1)

  • Figure 1: Random initial guess and outcomes of the interior-point method solving the optimization problem \ref{['eq:minimization_problem2']} when $m=2$. (A)-(B) starting feasible iterate and related solution given $r=10$. (C)-(D) starting feasible iterate and related solution given $r=20$.

Theorems & Definitions (43)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • ...and 33 more