Feed-anywhere ANN (I) Steady Discrete $\to$ Diffusing on Graph Hidden States

• URL: http://arxiv.org/abs/2507.20088v1
• Date: Sun, 27 Jul 2025 00:35:15 GMT
• Title: Feed-anywhere ANN (I) Steady Discrete $\to$ Diffusing on Graph Hidden States
• Authors: Dmitry Pasechnyuk-Vilensky, Daniil Doroshenko,
• Abstract summary: We propose a novel framework for learning hidden graph structures from data using geometric analysis and nonlinear dynamics.<n>Our model achieves stronger bounds than standard neural networks, with complexity dependent on the data manifold's topology.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a novel framework for learning hidden graph structures from data using geometric analysis and nonlinear dynamics. Our approach: (1) Defines discrete Sobolev spaces on graphs for scalar/vector fields, establishing key functional properties; (2) Introduces gauge-equivalent nonlinear Schr\"odinger and Landau--Lifshitz dynamics with provable stable stationary solutions smoothly dependent on input data and graph weights; (3) Develops a stochastic gradient algorithm over graph moduli spaces with sparsity regularization. Theoretically, we guarantee: topological correctness (homology recovery), metric convergence (Gromov--Hausdorff), and efficient search space utilization. Our dynamics-based model achieves stronger generalization bounds than standard neural networks, with complexity dependent on the data manifold's topology.

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