Persistent Nonnegative Matrix Factorization via Multi-Scale Graph Regularization

• URL: http://arxiv.org/abs/2602.22536v1
• Date: Thu, 26 Feb 2026 02:23:50 GMT
• Title: Persistent Nonnegative Matrix Factorization via Multi-Scale Graph Regularization
• Authors: Jichao Zhang, Ran Miao, Limin Li,
• Abstract summary: We propose persistent nonnegative matrix factorization (pNMF) that produces a sequence of persistence-aligned embeddings rather than a single one.<n>We analyze the structural properties of the embeddings along the scale parameter and establish bounds on their increments between consecutive scales.<n>The resulting model defines a nontrivial solution path across scales, rather than a single factorization, which poses new computational challenges.
• Score: 5.524295668575852
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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