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cs.LGmath.GNJennifer Rozenblit, Chenguang Yang, Yuxin Liu et al.
·
Oct 1, 2026
This paper tackles the tendency of graph diffusion models to lose global topology—such as cycles and higher-order motifs—during local denoising. It introduces MG-Diff, a framework that extracts a discrete Morse skeleton from a graph and conditions generation on this sparse topological backbone. By framing diffusion as traversing a discrete cobordism, the authors aim to anchor the generative trajectory to structurally critical events rather than treating topology as an afterthought. The promise is better preservation of connectivity and improved uncertainty quantification in spatio-temporal forecasting.
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.