Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study
Dmitry Pasechnyuk-Vilensky, Martin Takáč
TL;DR
The paper develops a geometric theory of neural networks where each layer is a fixed point of a dissipative Schrödinger-type dynamics on a learned latent graph. It establishes existence, uniqueness, and smooth dependence of equilibria, and shows a Bloch-map equivalence to Landau–Lifshitz dynamics, grounding the state evolution on a manifold. Training is formulated as optimization on a stratified moduli space of graphs with a Kaehler–Hessian metric, yielding convergence guarantees and geometry-aware generalization bounds that scale with edge count, degree, and Gromov–Hausdorff distortion. A key result is the equivalence between sequential stationary layers and a single global diffusion on a supra-graph, with backpropagation realized as the adjoint of the global stationary system; the framework extends to directed and vector-valued settings via sheaf Laplacians and unitary connections, unifying scalar, directed, and sheaf-based architectures. Together, these contributions provide a compact, interpretable, and analytically tractable foundation for learning latent graph geometry through fixed-point Schrödinger-type activations, with broad implications for causal inference and geometric deep learning.
Abstract
We develop a unified theoretical framework for neural architectures whose internal representations evolve as stationary states of dissipative Schrödinger-type dynamics on learned latent graphs. Each layer is defined by a fixed-point Schrödinger-type equation depending on a weighted Laplacian encoding latent geometry and a convex local potential. We prove existence, uniqueness, and smooth dependence of equilibria, and show that the dynamics are equivalent under the Bloch map to norm-preserving Landau--Lifshitz flows. Training over graph weights and topology is formulated as stochastic optimization on a stratified moduli space of graphs equipped with a natural Kähler--Hessian metric, ensuring convergence and differentiability across strata. We derive generalization bounds -- PAC-Bayes, stability, and Rademacher complexity -- in terms of geometric quantities such as edge count, maximal degree, and Gromov--Hausdorff distortion, establishing that sparsity and geometric regularity control capacity. Feed-forward composition of stationary layers is proven equivalent to a single global stationary diffusion on a supra-graph; backpropagation is its adjoint stationary system. Finally, directed and vector-valued extensions are represented as sheaf Laplacians with unitary connections, unifying scalar graph, directed, and sheaf-based architectures. The resulting model class provides a compact, geometrically interpretable, and analytically tractable foundation for learning latent graph geometry via fixed-point Schrödinger-type activations.