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

arXiv:2507.20088v4 Announce Type: replace Abstract: 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.

Abstract:We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-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 Kähler-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.

Machine Learning (cs.LG); Mathematical Physics (math-ph); Optimization and Control (math.OC); Machine Learning (stat.ML)

G.1.6; G.1.7; G.2.2

arXiv:2507.20088 [cs.LG]

(or arXiv:2507.20088v4 [cs.LG] for this version)

From: Dmitry A. Pasechnyuk [view email] [v1] Sun, 27 Jul 2025 00:35:15 UTC (35 KB) [v2] Fri, 7 Nov 2025 06:39:48 UTC (59 KB) [v3] Mon, 27 Apr 2026 13:26:22 UTC (69 KB) [v4] Mon, 7 Sep 2026 16:29:12 UTC (68 KB)

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

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