Non-negative Matrix Factorisation with Topological Regularisation

Abstract:We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

Machine Learning (cs.LG); Computational Geometry (cs.CG); Algebraic Topology (math.AT)

G.1.3; G.1.6; I.2.6; F.2.2

arXiv:2606.17531 [cs.LG]

(or arXiv:2606.17531v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.17531 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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