Abstract
Geometric Representation Learning (GRL) aims to approximate the non-Euclidean topology of high-dimensional data through discrete graph structures, grounded in the manifold hypothesis. However, traditional static graph construction methods based on Euclidean distance often fail to capture the intrinsic curvature characteristics of the data manifold. Although Ollivier-Ricci Curvature Flow (OCF) has proven to be a powerful tool for dynamic topological optimization, its core reliance on Optimal Transport (Wasserstein distance) leads to prohibitive computational complexity, severely limiting its application in large-scale datasets and deep learning frameworks. To break this bottleneck, this paper proposes a novel geometric evolution framework: Resistance Curvature Flow (RCF). Leveraging the concept of effective resistance from circuit physics, RCF transforms expensive curvature optimization into efficient matrix operations. This approach achieves over 100x computational acceleration while maintaining geometric optimization capabilities comparable to OCF. We provide an in-depth exploration of the theoretical foundations and dynamical principles of RCF, elucidating how it guides the redistribution of edge weights via curvature gradients to eliminate topological noise and strengthen local cluster structures. Furthermore, we provide a mechanistic explanation of RCF's role in manifold enhancement and noise suppression, as well as its compatibility with deep learning models. We design a graph optimization algorithm, DGSL-RCF, based on this framework. Experimental results across deep metric learning, manifold learning, and graph structure learning demonstrate that DGSL-RCF significantly improves representation quality and downstream task performance.
Abstract (translated)
几何表示学习（GRL）旨在通过离散图结构来近似高维数据的非欧几里得拓扑，这一过程基于流形假设。然而，传统的静态图构建方法通常依赖于欧氏距离，这种做法往往无法捕捉到数据流形内在的曲率特性。虽然奥利维尔-里奇曲率流动（OCF）已被证明是动态拓扑优化的强大工具，但其核心依赖于最优传输（Wasserstein距离），这导致了计算复杂度极高，严重限制了它在大规模数据集和深度学习框架中的应用。为了突破这一瓶颈，本文提出了一种新的几何演化框架：电阻曲率流动（RCF）。利用电路物理学中有效电阻的概念，RCF将昂贵的曲率优化转化为高效的矩阵运算。该方法实现了超过100倍的计算加速，并且在保持与OCF相当的几何优化能力的同时，还提供了更好的性能。 本文深入探讨了RCF的理论基础和动态原理，阐明了它是如何通过曲率梯度指导边权重的重新分配来消除拓扑噪声并强化局部集群结构的。此外，我们还从机制层面解释了RCF在流形增强和噪声抑制方面的作用及其与深度学习模型兼容性方面的优势。 基于这一框架，我们设计了一种图优化算法——DGSL-RCF（Deep Graph Structure Learning with Resistance Curvature Flow）。通过跨深度度量学习、流形学习以及图结构学习的实验结果表明，DGSL-RCF显著提高了表示质量和下游任务性能。
URL
https://arxiv.org/abs/2601.08149
