Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study

• URL: http://arxiv.org/abs/2507.20088v2
• Date: Fri, 07 Nov 2025 06:39:48 GMT
• Title: Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study
• Authors: Dmitry Pasechnyuk-Vilensky, Martin Takáč,
• Abstract summary: We develop a unified theoretical framework for neural architectures with internal representations evolving as stationary states of dissipative Schr"odinger-type dynamics on learned latent graphs.<n>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.<n>The resulting model class provides a compact, geometrically interpretable, and analytically tractable foundation for learning latent graph geometry via fixed-point Schr"odinger-type activations.
• Score: 1.1745324895296467
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We develop a unified theoretical framework for neural architectures whose internal representations evolve as stationary states of dissipative Schr\"odinger-type dynamics on learned latent graphs. Each layer is defined by a fixed-point Schr\"odinger-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 K\"{a}hler--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 Schr\"odinger-type activations.

Related papers

• Riemannian Liquid Spatio-Temporal Graph Network [6.583503277841693]
  Liquid-Constant networks (LTCs) excel at modeling irregularly-sampled dynamics but are fundamentally confined to Euclidean space.<n>This limitation introduces significant geometric distortion when representing real-world graphs with inherent non-Euclidean structures.<n>We introduce a framework that unifies continuous-time liquid dynamics with inductive inductive geometric biases.<n>RLSTG achieves superior performance on graphs with complex structures.
  arXiv  Detail & Related papers  (2026-01-20T16:09:05Z)
• The Neural Differential Manifold: An Architecture with Explicit Geometric Structure [8.201374511929538]
  This paper introduces the Neural Differential Manifold (NDM), a novel neural network architecture that explicitly incorporates geometric structure into its fundamental design.<n>We analyze the theoretical advantages of this approach, including its potential for more efficient optimization, enhanced continual learning, and applications in scientific discovery and controllable generative modeling.
  arXiv  Detail & Related papers  (2025-10-29T02:24:27Z)
• Adaptive Riemannian Graph Neural Networks [29.859977834688625]
  We introduce a novel framework that learns a continuous and anisotropic metric tensor field over the graph.<n>It allows each node to determine its optimal local geometry, enabling the model to fluidly adapt to the graph's structural landscape.<n>Our method demonstrates superior performance on both homophilic and heterophilic benchmark geometries.
  arXiv  Detail & Related papers  (2025-08-04T16:55:02Z)
• Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning [73.18052192964349]
  We develop a theoretical framework that explains how discrete symbolic structures can emerge naturally from continuous neural network training dynamics.<n>By lifting neural parameters to a measure space and modeling training as Wasserstein gradient flow, we show that under geometric constraints, the parameter measure $mu_t$ undergoes two concurrent phenomena.
  arXiv  Detail & Related papers  (2025-06-26T22:40:30Z)
• What Improves the Generalization of Graph Transformers? A Theoretical Dive into the Self-attention and Positional Encoding [67.59552859593985]
  Graph Transformers, which incorporate self-attention and positional encoding, have emerged as a powerful architecture for various graph learning tasks. This paper introduces first theoretical investigation of a shallow Graph Transformer for semi-supervised classification.
  arXiv  Detail & Related papers  (2024-06-04T05:30:16Z)
• LSEnet: Lorentz Structural Entropy Neural Network for Deep Graph Clustering [59.89626219328127]
  Graph clustering is a fundamental problem in machine learning. Deep learning methods achieve the state-of-the-art results in recent years, but they still cannot work without predefined cluster numbers. We propose to address this problem from a fresh perspective of graph information theory.
  arXiv  Detail & Related papers  (2024-05-20T05:46:41Z)
• Topological Obstructions and How to Avoid Them [22.45861345237023]
  We show that local optima can arise due to singularities or an incorrect degree or winding number. We propose a new flow-based model that maps data points to multimodal distributions over geometric spaces.
  arXiv  Detail & Related papers  (2023-12-12T18:56:14Z)
• Improving embedding of graphs with missing data by soft manifolds [51.425411400683565]
  The reliability of graph embeddings depends on how much the geometry of the continuous space matches the graph structure. We introduce a new class of manifold, named soft manifold, that can solve this situation. Using soft manifold for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets.
  arXiv  Detail & Related papers  (2023-11-29T12:48:33Z)
• Supercharging Graph Transformers with Advective Diffusion [28.40109111316014]
  This paper proposes Advective Diffusion Transformer (AdvDIFFormer), a physics-inspired graph Transformer model designed to address this challenge.<n>We show that AdvDIFFormer has provable capability for controlling generalization error with topological shifts.<n> Empirically, the model demonstrates superiority in various predictive tasks across information networks, molecular screening and protein interactions.
  arXiv  Detail & Related papers  (2023-10-10T08:40:47Z)
• Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning [77.1421343649344]
  We propose a generalization of Transformers towards operating entirely on the product of constant curvature spaces. We also provide a kernelized approach to non-Euclidean attention, which enables our model to run in time and memory cost linear to the number of nodes and edges.
  arXiv  Detail & Related papers  (2023-09-08T02:44:37Z)
• Latent Graph Inference using Product Manifolds [0.0]
  We generalize the discrete Differentiable Graph Module (dDGM) for latent graph learning. Our novel approach is tested on a wide range of datasets, and outperforms the original dDGM model.
  arXiv  Detail & Related papers  (2022-11-26T22:13:06Z)
• Unveiling the Sampling Density in Non-Uniform Geometric Graphs [69.93864101024639]
  We consider graphs as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius. In a social network communities can be modeled as densely sampled areas, and hubs as nodes with larger neighborhood radius. We develop methods to estimate the unknown sampling density in a self-supervised fashion.
  arXiv  Detail & Related papers  (2022-10-15T08:01:08Z)
• Holographic properties of superposed quantum geometries [0.0]
  We study the holographic properties of a class of quantum geometry states characterized by a superposition of discrete geometric data. This class includes spin networks, the kinematic states of lattice gauge theory and discrete quantum gravity.
  arXiv  Detail & Related papers  (2022-07-15T17:37:47Z)
• Optimal Propagation for Graph Neural Networks [51.08426265813481]
  We propose a bi-level optimization approach for learning the optimal graph structure. We also explore a low-rank approximation model for further reducing the time complexity.
  arXiv  Detail & Related papers  (2022-05-06T03:37:00Z)
• Learning to Learn Graph Topologies [27.782971146122218]
  We learn a mapping from node data to the graph structure based on the idea of learning to optimise (L2O) The model is trained in an end-to-end fashion with pairs of node data and graph samples. Experiments on both synthetic and real-world data demonstrate that our model is more efficient than classic iterative algorithms in learning a graph with specific topological properties.
  arXiv  Detail & Related papers  (2021-10-19T08:42:38Z)
• Regularization of Mixture Models for Robust Principal Graph Learning [0.0]
  A regularized version of Mixture Models is proposed to learn a principal graph from a distribution of $D$-dimensional data points. Parameters of the model are iteratively estimated through an Expectation-Maximization procedure.
  arXiv  Detail & Related papers  (2021-06-16T18:00:02Z)
• Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphs [77.33781731432163]
  We learn dynamic graph representation in hyperbolic space, for the first time, which aims to infer node representations. We present a novel Hyperbolic Variational Graph Network, referred to as HVGNN. In particular, to model the dynamics, we introduce a Temporal GNN (TGNN) based on a theoretically grounded time encoding approach.
  arXiv  Detail & Related papers  (2021-04-06T01:44:15Z)
• Hyperbolic Graph Embedding with Enhanced Semi-Implicit Variational Inference [48.63194907060615]
  We build off of semi-implicit graph variational auto-encoders to capture higher-order statistics in a low-dimensional graph latent representation. We incorporate hyperbolic geometry in the latent space through a Poincare embedding to efficiently represent graphs exhibiting hierarchical structure.
  arXiv  Detail & Related papers  (2020-10-31T05:48:34Z)
• Block-Approximated Exponential Random Graphs [77.4792558024487]
  An important challenge in the field of exponential random graphs (ERGs) is the fitting of non-trivial ERGs on large graphs. We propose an approximative framework to such non-trivial ERGs that result in dyadic independence (i.e., edge independent) distributions. Our methods are scalable to sparse graphs consisting of millions of nodes.
  arXiv  Detail & Related papers  (2020-02-14T11:42:16Z)
• Understanding Graph Neural Networks with Generalized Geometric Scattering Transforms [67.88675386638043]
  The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. We introduce windowed and non-windowed geometric scattering transforms for graphs based upon a very general class of asymmetric wavelets. We show that these asymmetric graph scattering transforms have many of the same theoretical guarantees as their symmetric counterparts.
  arXiv  Detail & Related papers  (2019-11-14T17:23:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.