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This paper advances machine learning by integrating persistent homology into non-negative matrix factorisation (NMF) to enforce topological constraints on learned basis functions. The method offers a threshold-free approach that improves the interpretability of data representations across images, time-series, and graph signals without relying on discrete approximations. For researchers in computational geometry and unsupervised learning, this framework provides a robust tool for modeling structured domains where intrinsic topology dictates feature quality.
Non-negative Matrix Factorisation with Topological Regularisation
arXiv:2606.17531v1 Announce Type: new 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.
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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