# Geometry-Aware Operator Families for Structured Representation Learning

arXiv: https://arxiv.org/abs/2609.33089 · HTML: https://mlsanity.com/paper/2609.33089/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 54 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Filling the G_ap_s: Multivariate Time Series Imputation by Graph Neural Networks](https://mlsanity.com/paper/2108.00298/) (2021)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting](https://mlsanity.com/paper/1707.01926/) (2017)
- [Geometry of Drifting MDPs with Path-Integral Stability Certificates](https://mlsanity.com/paper/2601.21991/) (2026)
- [Manifold-Constrained Energy-Based Transition Models for Offline Reinforcement Learning](https://mlsanity.com/paper/2602.02900/) (2026)
- [Operator-Guided Invariance Learning for Continuous Reinforcement Learning](https://mlsanity.com/paper/2605.06500/) (2026)
- [Metric-Gradient Projection for Stable Multi-Agent Policy Learning](https://mlsanity.com/paper/2605.18809/) (2026)
- [Matrix-Space Reinforcement Learning for Reusing Local Transition Geometry](https://mlsanity.com/paper/2605.14304/) (2026)
- [Minimal Markovization via Stable Quotients in Holonomy-Cover Decision Processes](https://mlsanity.com/paper/2607.27132/) (2026)
- [Learning Not to Optimize: Physics-Informed Action-Space Reshaping for Intent-Based Network Control](https://mlsanity.com/paper/2608.00908/) (2026)
- [Differentiable Policy Transport over Multi-Layer Network Feasibility Geometry](https://mlsanity.com/paper/2609.26068/) (2026)

## Abstract (arXiv)

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 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, and oversmoothing behavior, while controlled experiments validate these mechanisms and experiments on PEMS-BAY and METR-LA achieve the lowest mean MAE across all reported regional-outage settings, improving over the strongest retained baseline by 2.4%--8.8% at 30% missing sensors.