# Learning Latent Graph Geometry via Fixed-Point Schr"odinger-Type Activation: A Theoretical Study

arXiv: https://arxiv.org/abs/2507.20088 · HTML: https://mlsanity.com/paper/2507.20088/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 98 references that are in the map.
- First posted on arXiv 27 Jul 2025.

## Builds on (most cited first, 18 of 18)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Sheaf theory: from deep geometry to deep learning](https://mlsanity.com/paper/2502.15476/) (2025)

## Abstract (arXiv)

We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schr"odinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate K"ahler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.