Graph Representation via Elements of Discrete Morse and Cobordism Theories

2026-10-01

AI Summary

This study addresses the performance bottlenecks of graph diffusion models in spatiotemporal prediction and generation tasks, as well as the limitations of topological methods in machine learning. For the first time, it systematically introduces discrete Morse theory and cobordism theory from pure mathematics into graph diffusion models. By constructing the MG-Diff pipeline, this work achieves structural optimization of the graph diffusion process using low-dimensional topological tools and establishes rigorous theoretical stability guarantees. Experimental results demonstrate that the proposed approach significantly enhances graph prediction and generation performance, validating the effectiveness of low-dimensional topology in augmenting the representational capacity of diffusion models. Ultimately, this research pioneers a new paradigm for interdisciplinary studies at the intersection of topology and machine learning.

Abstract

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.

Problem

• Graph Representation
• Discrete Morse Theory
• Cobordism Theory
• Graph Diffusion Models
• Low-dimensional Topology

Innovation

• Discrete Morse Theory
• Cobordism Theory
• Graph Diffusion Models
• Topological Machine Learning
• Spatio-temporal Graph Forecasting