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.

翻译：我们研究通过正则化基函数的拓扑结构来学习非负矩阵分解（NMF）中的可解释基。该方法源于一个观察：许多数据模态可被视为结构化域上的非负函数，此时基的质量内在地与其拓扑性质相关。然而，将支撑集拓扑结构纳入考量的朴素方法常受离散性和阈值依赖性的阻碍，无法适用于连续优化。我们通过采用持续同调作为稳定且无阈值的拓扑量化器，并设计可整合到NMF目标函数中作为正则项的拓扑评分，解决了这些挑战。由此产生的框架能够统一建模空间连贯的图像组件、周期性时间序列结构以及类团图信号。