Persistent Nonnegative Matrix Factorization via Multi-Scale Graph Regularization

TL;DR

This paper introduces persistent nonnegative matrix factorization (pNMF), a multi-scale approach that captures evolving connectivity structures in data using persistent homology, providing more insightful embeddings than traditional single-scale NMF.

Contribution

The paper proposes a novel multi-scale NMF framework leveraging persistent homology to identify canonical scales and enforce cross-scale consistency, advancing interpretability and robustness of embeddings.

Findings

Effective multi-scale embeddings demonstrated on synthetic data.

Improved analysis of single-cell RNA sequencing data.

Guaranteed convergence of the optimization algorithm.

Abstract

Matrix factorization techniques, especially Nonnegative Matrix Factorization (NMF), have been widely used for dimensionality reduction and interpretable data representation. However, existing NMF-based methods are inherently single-scale and fail to capture the evolution of connectivity structures across resolutions. In this work, we propose persistent nonnegative matrix factorization (pNMF), a scale-parameterized family of NMF problems, that produces a sequence of persistence-aligned embeddings rather than a single one. By leveraging persistent homology, we identify a canonical minimal sufficient scale set at which the underlying connectivity undergoes qualitative changes. These canonical scales induce a sequence of graph Laplacians, leading to a coupled NMF formulation with scale-wise geometric regularization and explicit cross-scale consistency constraint. We analyze the structural properties of the embeddings along the scale parameter and establish bounds on their increments between consecutive scales. The resulting model defines a nontrivial solution path across scales, rather than a single factorization, which poses new computational challenges. We develop a sequential alternating optimization algorithm with guaranteed convergence. Numerical experiments on synthetic and single-cell RNA sequencing datasets demonstrate the effectiveness of the proposed approach in multi-scale low-rank embeddings.