🤖 AI Summary
This work addresses the challenge in non-negative matrix factorization (NMF) of simultaneously achieving interpretable basis functions and desirable topological structure, a limitation often exacerbated by reliance on discrete formulations or thresholding. To overcome this, the study introduces persistent homology into NMF for the first time, proposing a threshold-free, continuously optimized topological regularization framework. By embedding a topological loss term derived from persistent homology directly into the objective function, the method guides the learning of non-negative basis functions that exhibit specific topological features—such as spatial coherence, periodicity, or clique-like structures. The approach is unified across diverse data modalities, including images, time series, and graph signals, significantly enhancing both the structural plausibility and interpretability of the resulting decompositions.
📝 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.