New NMF Method Incorporates Data Topology for Interpretable Bases.

Non-negative Matrix Factorisation (NMF) is a widely used technique for decomposing data into interpretable components. However, learning basis functions that accurately reflect the underlying structure or "topology" of the data has been challenging, especially when dealing with data modalities that can be viewed as non-negative functions on structured domains. Traditional methods often struggle with issues like discreteness and dependence on arbitrary thresholds. This paper introduces a novel NMF framework that overcomes these limitations by incorporating topological regularisation. The core idea is to use persistent homology, a robust and threshold-free method for quantifying topological features, to design scores that can be integrated directly into the NMF objective function. This allows the model to learn basis functions that are topologically consistent with the data. The resulting framework offers a unified way to model various data structures, including spatially coherent image components, periodic time-series patterns, and clique-like graph signals. By explicitly regularizing the topology of the learned bases, the method aims to produce more meaningful and interpretable decompositions across diverse applications.