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This research introduces Top-NMF, a novel framework that integrates persistent homology into non-negative matrix factorization to enforce interpretable structural priors like connected components and loops. By relaxing discrete topological conditions into continuous optimization scores, the method improves basis learning for images, graphs, and time-series data without over-specifying component forms. The work advances machine learning theory by demonstrating how algebraic topology can stabilize representation learning and enhance atom recovery in structured domains.
Non-negative Matrix Factorisation with Topological Regularisation
arXiv:2606.17531v2 Announce Type: replace Abstract: Non-negative matrix factorisation (NMF) learns additive representations from data, but non-negativity alone does not ensure interpretable basis functions. We introduce Top-NMF, which guides basis learning through general topological preferences without prescribing the detailed form of the components. Observations and basis vectors are treated as non-negative functions on structured domains. Discrete topological conditions are difficult to optimise directly. Our guiding viewpoint is that persistent homology provides a stable, continuous relaxation of ordinary homological invariants: it tracks connected components and loops across thresholds, allowing structural preferences to be expressed through continuous scores suitable for optimisation. We construct such scores for connected image parts, clique-like graph structure, and periodic time-series components, and incorporate them into a common NMF objective. For graph data, we derive an exact maximum-weight-spanning-tree formula and characterise all local minimisers of the score, showing that their positive-weight edges form disjoint cliques. We establish the regularity of the scores, describe their derivatives, and analyse projected first-order optimisation. Numerical studies on synthetic and real data demonstrate how these priors guide basis learning and examine the balance between structural agreement, atom recovery, and reconstruction accuracy.
Abstract:Non-negative matrix factorisation (NMF) learns additive representations from data, but non-negativity alone does not ensure interpretable basis functions. We introduce Top-NMF, which guides basis learning through general topological preferences without prescribing the detailed form of the components. Observations and basis vectors are treated as non-negative functions on structured domains. Discrete topological conditions are difficult to optimise directly. Our guiding viewpoint is that persistent homology provides a stable, continuous relaxation of ordinary homological invariants: it tracks connected components and loops across thresholds, allowing structural preferences to be expressed through continuous scores suitable for optimisation. We construct such scores for connected image parts, clique-like graph structure, and periodic time-series components, and incorporate them into a common NMF objective. For graph data, we derive an exact maximum-weight-spanning-tree formula and characterise all local minimisers of the score, showing that their positive-weight edges form disjoint cliques. We establish the regularity of the scores, describe their derivatives, and analyse projected first-order optimisation. Numerical studies on synthetic and real data demonstrate how these priors guide basis learning and examine the balance between structural agreement, atom recovery, and reconstruction accuracy.
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:2606.17531v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.17531 Focus to learn more arXiv-issued DOI via DataCite
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