Persistent Nonnegative Matrix Factorization via Multi-Scale Graph Regularization

2026-02-25

AI Summary

This work proposes persistent Non-negative Matrix Factorization (pNMF), a novel extension of traditional NMF that addresses its limitation to a single scale by incorporating persistent homology to capture structural evolution across multiple resolutions. By parameterizing the decomposition over a scale parameter, pNMF identifies critical topological scales and constructs graph Laplacian regularizers at these scales to enforce cross-scale consistency in the low-rank embedding. Theoretical analysis provides bounds on the variation of embeddings between adjacent scales. Experiments on both synthetic and single-cell RNA sequencing data demonstrate that pNMF yields geometrically coherent, multi-scale embeddings that are biologically interpretable, effectively revealing underlying data structures that vary across resolutions.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Nonnegative Matrix Factorization
• multi-scale
• persistent homology
• graph regularization
• connectivity structure

Innovation

• persistent homology
• multi-scale graph regularization
• nonnegative matrix factorization
• scale-wise consistency
• persistence-aligned embeddings