发表机构
University of Texas, Austin; University of California, Riverside; Virginia Tech（德克萨斯大学奥斯汀分校; 加利福尼亚大学河滨分校; 弗吉尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出利用离散莫尔斯理论和协边理论工具，通过MG-Diff流程增强图扩散模型，提供理论稳定性保证，并展示其在时空图预测和图再生中的有效性。
AI 中文摘要
拓扑学，就其本质和设计而言，适用于非线性、多尺度和非平稳的结构——然而，在机器学习中，其应用仍主要局限于拓扑数据分析。我们主张，低维拓扑学中几乎完全局限于纯数学领域（如莫尔斯理论）的工具，为数据生成过程及其上构建的学习任务的隐藏结构提供了一个强大、互补且几乎未被探索的视角。在此，我们引入协边理论的概念，并利用离散莫尔斯理论的工具，通过我们的流程MG-Diff来提升图扩散模型的性能。此外，我们推导了理论保证和充分条件，使得在正决策差距下，莫尔斯理论工具及其在诱导扩散引导中的应用在小扰动下保持稳定。最后，我们展示了离散莫尔斯理论在图扩散模型中用于时空图预测和图再生的实用性，并论证这些应用仅是低维拓扑学能为机器学习领域提供的一小部分。
英文摘要
Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.
