用拓扑正则化提升非负矩阵分解的可解释性
Non-negative Matrix Factorisation with Topological Regularisation
- 引入持久同调作为无阈值拓扑度量，优化基函数结构
- 在图像、时序和图信号中实现空间连贯与周期性特征提取
- 适合关注数据结构解释性的研究者使用
我们通过正则化学习到的基函数的拓扑结构，研究非负矩阵分解（NMF）中可解释基的学习。许多数据模态可视为结构化域上的非负函数，其基的质量与拓扑密切相关。然而，传统拓扑引入方法受离散性和阈值依赖限制，难以用于连续优化。为此，我们采用持久同调作为稳定、无阈值的拓扑量化工具，并设计可融入NMF目标函数的拓扑评分作为正则项。该框架统一建模了空间连贯的图像成分、周期性时间序列结构以及类团图信号。
原文摘要 · Abstract (English)
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.
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