An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

2026-01-07

AI Summary

This work addresses the limitation of existing linear GENEO representation theorems, which are restricted to homogeneous data spaces and thus unable to model symmetries across heterogeneous data. By introducing a generalized T-permutation measure, we establish—for the first time—a linear GENEO algebraic representation theorem applicable to heterogeneous perception pairs, thereby unifying geometric and topological deep learning frameworks. Leveraging group actions, equivariance, and functional analysis, we fully characterize the structure of linear GENEOs over heterogeneous spaces and prove that their operator space is compact and convex. Integrating this theory into an autoencoder architecture yields significant performance gains in experiments, demonstrating both the theoretical extensibility and practical utility of GENEOs in modern machine learning.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• GENEOs
• heterogeneous data spaces
• linear operators
• equivariance
• representation theorem

Innovation

• GENEOs
• representation theorem
• heterogeneous data spaces
• group equivariance
• T-permutant measures