Geometry-Aware Operator Families for Structured Representation Learning

Zuyuan Zhang, Fei Xu Yu, Tian Lan

The geometry of latent representations governs which components should interact and how information should propagate, making geometry-aware operator design a fundamental ingredient of structured deep representation learning. However, existing neural architectures typically rely on generic operator templates or geometry-specific constructions, creating a need for a unified framework that can derive admissible operators directly from fixed structural information while remaining adaptive to changing contexts. We introduce \emph{Geometry-Induced Operator Families} (GIOF), a general framework that converts fixed geometry into a structured family of propagation operators and dynamically selects an appropriate member of this family according to the current context. GIOF first transforms geometry-derived interaction channels into reusable generator bases, then combines them through a context-dependent selector and adaptive propagation scale, and finally realizes the selected operator through stable continuous-time propagation and a bottleneck residual layer. We establish theoretical guarantees covering parameter compression, identifiability, stability, locality, compositional structure,…