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.

翻译：几何与拓扑深度学习是快速发展的研究领域，通过利用几何与拓扑结构来增强机器学习能力。在此框架下，群等变非扩张算子（GENEOs）已成为编码对称性和设计高效、可解释神经架构的强大算子类别。GENEOs最初在拓扑数据分析中被提出，随后在深度学习中作为构建具有降低参数复杂度的等变模型的工具得到应用。GENEOs提供了一个连接几何与拓扑深度学习的统一框架，并将计算持续性图的算子作为特例包含其中。其理论基础依赖于群作用、等变性及算子空间的紧致性，使其植根于代数与几何的同时兼具数学严谨性与实际相关性。虽然先前的表示定理刻画了作用于同类数据空间的线性GENEOs，但许多实际应用需要处理异构数据空间之间的算子。本研究通过引入基于广义T置换测度的新型表示定理，解决了作用于不同感知对之间的线性GENEOs的表征问题。在数据域和群作用满足温和假设的条件下，我们的结果为此类算子提供了完整表征。我们进一步证明了线性GENEOs空间的紧致性与凸性。通过将所提框架应用于提升自编码器性能，我们展示了该理论的实际影响，凸显了GENEOs在现代机器学习应用中的重要性。