Non-negative Matrix Factorisation with Topological Regularisation
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.