Persistent Nonnegative Matrix Factorization via Multi-Scale Graph Regularization
arXiv:2602.22536v1 Announce Type: new 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 paramete
arXiv:2602.22536v1 Announce Type: new 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.
Executive Summary
This article proposes a novel approach to Nonnegative Matrix Factorization (NMF) called persistent nonnegative matrix factorization (pNMF), which leverages persistent homology to identify canonical scales in the underlying connectivity structure. By inducing a sequence of graph Laplacians, pNMF formulates a coupled NMF problem with scale-wise geometric regularization and explicit cross-scale consistency constraints. The resulting model defines a nontrivial solution path across scales, posing new computational challenges. A sequential alternating optimization algorithm with guaranteed convergence is developed. Numerical experiments demonstrate the effectiveness of pNMF in multi-scale low-rank embeddings. The approach has the potential to capture the evolution of connectivity structures across resolutions, making it particularly relevant to applications where scale is a crucial factor.
Key Points
- ▸ pNMF leverages persistent homology to identify canonical scales in the underlying connectivity structure.
- ▸ pNMF formulates a coupled NMF problem with scale-wise geometric regularization and explicit cross-scale consistency constraints.
- ▸ A sequential alternating optimization algorithm with guaranteed convergence is developed for pNMF.
Merits
Strength in Scale
pNMF's ability to capture the evolution of connectivity structures across resolutions is a significant strength, making it particularly relevant to applications where scale is a crucial factor.
Computational Efficiency
The development of a sequential alternating optimization algorithm with guaranteed convergence ensures that pNMF is computationally efficient and scalable.
Demerits
Computational Complexity
The introduction of scale-wise geometric regularization and explicit cross-scale consistency constraints increases the computational complexity of pNMF, making it challenging to deploy in large-scale applications.
Interpretability
The nontrivial solution path across scales may make it challenging to interpret the results of pNMF, particularly for users without a strong background in persistent homology and graph regularization.
Expert Commentary
The work on pNMF is a significant contribution to the field of nonnegative matrix factorization, and demonstrates the potential of persistent homology to improve the accuracy and interpretability of NMF-based methods. However, the introduction of scale-wise geometric regularization and explicit cross-scale consistency constraints increases the computational complexity of pNMF, making it challenging to deploy in large-scale applications. Nevertheless, the development of a sequential alternating optimization algorithm with guaranteed convergence ensures that pNMF is computationally efficient and scalable. In conclusion, pNMF has the potential to be a powerful tool for multi-scale data analysis, and its implications for a wide range of applications are significant.
Recommendations
- ✓ Future work should focus on developing more efficient algorithms for pNMF, particularly for large-scale applications.
- ✓ The use of pNMF should be explored in a wide range of applications, including image and video processing, natural language processing, and bioinformatics.
