An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

TL;DR

This work generalizes the algebraic representation of Group Equivariant Non-Expansive Operators (GENEOs) from linear operators acting on the same data space to those between heterogeneous perception pairs by introducing generalized $T$-permutant measures. It proves a complete representation: a linear GEO $(F,T)$ is characterized by a $T$-permutant measure $\mu$ via $F(\varphi)=\sum_{h\in X^Y} \varphi h\ \mu(h)$, with transitivity of $T(G)$ on $Y$ ensuring existence; for GENEOs, the extra condition $\sum_{h} |\mu(h)|\le 1$ governs non-expansivity, and a precise norm relation holds. The paper further shows that the space of linear GENEOs is compact and convex, expressible as a convex hull of normalized generators, and demonstrates a practical autoencoder enhancement by inserting a GENEO layer before reconstruction on MNIST, yielding improved robustness to noise. Overall, the results bridge algebraic geometry and geometric/topological DL, providing both theoretical foundations and a tangible ML application for GENEOs.

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.

Figures (2)

• Figure 1: Left: CNN classification accuracy on reconstructed images. Right: Mean Absolute Error (MAE) between reconstructed and clean images as a complementary measure. The GENEO CAE achieves both the highest classification accuracy and, with the sole exception of 40% noise, the lowest reconstruction error.
• Figure 2: Reconstruction results for test images ID 400 (left) and 500 (right) across increasing salt & pepper noise levels. Columns show original images and reconstructions from AE, CAE, VAE, and GENEO CAE. The GENEO CAE demonstrates superior ability to preserve digit structure under heavy noise corruption.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Definition 3.1
• Proposition 3.2
• proof
• Theorem 4.1
• Lemma 4.2
• proof