Non-negative Matrix Factorisation with Topological Regularisation
For practitioners needing interpretable decompositions in structured data, this provides a unified topological regularisation framework, though it is an incremental extension of existing NMF and topological methods.
The paper introduces a topological regularisation for NMF using persistent homology to learn interpretable bases with desired topological features, demonstrating improved interpretability on image, time-series, and graph data.
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.