Graph diffusion performance improved using low-dimensional topology tools
Graph Representation via Elements of Discrete Morse and Cobordism Theories
Machine Learning
Summary
Graphs are tricky because their data is complex and can change in many ways. The authors use special ideas from a branch of math called topology, which studies shapes and spaces, to better understand and work with graphs. They combine concepts from discrete Morse theory and cobordism theory to help improve graph data processing. Their approach helps make certain graph-based predictions more stable even when the data has small changes. They also show their method works for predicting how things change over time and for recreating graph structures.
What this means in practice
- •For data scientists: Improve accuracy and stability in predicting changes in spatio-temporal graph data using topological methods.
- •For graph algorithm engineers: Enhance graph regeneration tasks by guiding diffusion models with discrete Morse theory for better structural recovery.
Authors
Jennifer Rozenblit, Chenguang Yang, Yuxin Liu, Yuzhou Chen, Yulia Gel
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.