An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

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.

• The paper presents an algebraic representation theorem that characterizes all linear GENEOs using generalized T-permutant measures.
• It decomposes operators into convex sums of permutation matrices, revealing the compact and convex structure of the GENEO space.
• Experimental results on noise-robust autoencoders demonstrate that GENEO-enhanced architectures outperform traditional models in reconstruction fidelity.

Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

Overview of GENEOs and Geometric/Topological Foundations

Group Equivariant Non-Expansive Operators (GENEOs) constitute a class of linear operators central to the intersection of geometric and topological deep learning. By virtue of their equivariance properties with respect to group actions and non-expansivity with respect to selected norms, GENEOs provide interpretable, structure-preserving mechanisms that encode symmetries directly into neural architectures. The foundational premise of GENEOs is their ability to unify the geometric priors pervasive in geometric deep learning [Br17, BrBr21, bergomi2019towards] with the global descriptors of topological data analysis, such as persistence diagrams [FrJa16, Zia2024]. Notable is their application in constructing neural networks with drastically reduced parameter counts while maintaining robustness and interpretability, by systematically leveraging symmetry and equivariance.

The linkage to topological deep learning is especially manifested when GENEOs subsume classical TDA operators as special cases, e.g., the computation of persistence diagrams is formulated as a GENEO in this framework. The mathematical formalism is built upon perception pairs (data spaces with specific group actions), group homomorphisms, and the algebraic structure of non-expansive maps, notably in finite domains.

Generalized Permutant Measures and the Representation Theorem

A critical advance proposed in the paper is the extension of previous representation theorems that required homogeneous data domains. Real-world applications, however, necessitate equivariant operators between heterogeneous perception pairs. The central result is the characterization of all linear GENEOs between finite perception pairs under transitive group actions, via generalized $T$-permutant measures.

Given finite sets $X$, $Y$, subgroups $G \subseteq \operatorname{Aut}(X)$, $K \subseteq \operatorname{Aut}(Y)$, and a group homomorphism $T: G \to K$, every linear GENEO $(F, T): (R^X, G) \to (R^Y, K)$ admits a decomposition

$F(\varphi) = \sum_{h \in X^Y} \varphi h\ \mu(h)$

where $\mu$ is a generalized $T$-permutant measure (i.e., a measure invariant under the induced group action on $X$0 by $X$1), and the non-expansivity condition is enforced by requiring $X$2. Notably, the identification of GENEOs as sums weighted by generalized permutant measures yields explicit, constructive parameterizations in terms of the group-theoretic structure of the data spaces.

The proof employs a decomposition of linear operators into convex combinations of rectangular permutation matrices, exploiting the convex structure and the orbit decomposition induced by group actions. Key intermediary results guarantee row-permutation structure in associated matrices and establish mutual singularity of the positive/negative Hahn-Jordan components of measures.

Compactness and Convexity of the GENEO Space

The article rigorously analyzes the topological and geometric properties of the GENEO space. For fixed group homomorphism $X$3, the set $X$4 of all linear GENEOs is shown to be a compact, convex polytope in the operator norm topology, generated by the convex hull of explicit canonical operators associated with the orbits of the induced $X$5 action. This provides precise criteria for the construction of the GENEO operator set and further assures computational tractability for model selection and search [Ah26, FaFeFr2023].

Effect on Robustness: Application to Autoencoders

The utility of the representation theorem is realized experimentally by application to noise-robust autoencoders. By introducing a preliminary GENEO layer—constructed via explicit equivariant linear maps parameterized by translation symmetry on the discrete torus—the architecture is shown to have marked advantages in preserving digit structure under high salt-and-pepper noise on MNIST images.

Across a range of models (standard autoencoder, convolutional autoencoder, variational autoencoder, and GENEO-enhanced convolutional autoencoder), quantitative results substantiate the claims:

• Classification Accuracy and MAE: The GENEO CAE consistently achieves the highest post-reconstruction CNN accuracy and—at almost all noise levels—the lowest mean absolute error (MAE) (Figure 1).

Figure 1: Left: CNN classification accuracy on reconstructed MNIST digits under various noise levels; right: MAE between reconstructed and clean images. GENEO CAE exhibits the best robustness at all levels except the highest noise.

• Qualitative Reconstruction: Visual evidence further highlights the capacity of GENEO operators to preserve digit topology and semantics even under severe corruption, outperforming both baseline and convolutional variants (Figure 2).

Figure 2: Reconstructions for representative test samples at increasing noise, comparing AE, CAE, VAE, and GENEO CAE. GENEO CAE preserves structural integrity under heavy corruption.

These results underscore the practical impact of operator-theoretic symmetry encoding, extending recent work on topological autoencoders [pmlr-v119-moor20a], and demonstrate that GENEOs can be systematically engineered to maximize robustness.

Theoretical and Practical Implications

The algebraic characterization and constructive representation of GENEOs substantially broaden the theoretical landscape of geometric machine learning. By allowing equivariant, interpretable, and parameter-efficient architectures tailored to the symmetry group of the underlying data, the work paves the way for principled design of learning systems resilient to noise, deformation, and invariance transformations. GENEO-based models, such as SCENE-Net [LaMiBoFrSo25] and GENEO-Net [BoFrMi25], have already demonstrated state-of-the-art performance in low-data and explainable ML settings, validating the underlying framework.

Future directions include removing finiteness constraints on data domains (thus extending the representation theorem to infinite spaces and functional data), automating the algorithmic construction of permutant measures, and integrating GENEOs in broader ML pipelines beyond encoding/decoding—such as in graph analysis [BoFeFr25], protein binding pocket detection [BoFrMi25], and explainable AI [CoBoGiGiPeFr25]. The robustness and compactness results also suggest tractable optimization landscapes for training and model selection.

Conclusion

This paper rigorously establishes a general representation theorem for linear GENEOs between finite perception pairs equipped with transitive group actions, parameterized via generalized $X$6-permutant measures. The explicit algebraic characterization leads to a compact and convex description of the GENEO space and confers robust practical benefits in noise-tolerant learning, as evidenced by gains in autoencoder reconstruction fidelity. The framework seamlessly bridges foundational algebraic, geometric, and topological ML theory with engineered operator construction, and solidifies GENEOs as indispensable tools in the design of interpretable and resilient learning systems.