An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

TL;DR

This paper introduces a new algebraic representation theorem for linear GENEOs acting between different data spaces, enhancing the theoretical foundation and practical applications in geometric deep learning.

Contribution

It extends the existing theory of GENEOs by characterizing operators between heterogeneous data spaces using generalized T-permutant measures.

Findings

Complete characterization of linear GENEOs between different perception pairs.

Proved compactness and convexity of the space of linear GENEOs.

Demonstrated improved autoencoder performance using the new framework.

Abstract

Geometric and Topological Deep Learning are rapidly growing research areas that enhance machine learning through the use of geometric and topological structures. Within this framework, Group Equivariant Non-Expansive Operators (GENEOs) have emerged as a powerful class of operators for encoding symmetries and designing efficient, interpretable neural architectures. Originally introduced in Topological Data Analysis, GENEOs have since found applications in Deep Learning as tools for constructing equivariant models with reduced parameter complexity. GENEOs provide a unifying framework bridging Geometric and Topological Deep Learning and include the operator computing persistence diagrams as a special case. Their theoretical foundations rely on group actions, equivariance, and compactness properties of operator spaces, grounding them in algebra and geometry while enabling both mathematical rigor and practical relevance. While a previous representation theorem characterized linear GENEOs acting on data of the same type, many real-world applications require operators between heterogeneous data spaces. In this work, we address this limitation by introducing a new representation theorem for linear GENEOs acting between different perception pairs, based on generalized T-permutant measures. Under mild assumptions on the data domains and group actions, our result provides a complete characterization of such operators. We also prove the compactness and convexity of the space of linear GENEOs. We further demonstrate the practical impact of this theory by applying the proposed framework to improve the performance of autoencoders, highlighting the relevance of GENEOs in modern machine learning applications.