Graph Regularized Deep Symmetric Nonnegative Matrix Factorization

doi:10.1016/j.knosys.2026.116556

Graph Regularized Deep Symmetric Nonnegative Matrix Factorization

2026-07-01

0

PRE

AI

S

Saeed Karami *

F

Farid Saberi-Movahed *

DOI：10.1016/j.knosys.2026.116556

Original

Original request for help

Save

Abstract

En 中文

Symmetric Nonnegative Matrix Factorization (SNMF) has shown strong potential for clustering data with linear and nonlinear structures by factorizing a symmetric similarity matrix into a clustering assignment matrix and its transpose. However, existing SNMF-based methods face two important limitations. First, most of them are shallow and thus cannot sufficiently capture hierarchical and multi-level structures in complex data. Second, although graph-regularized SNMF methods preserve local geometry, they usually impose graph constraints on a single representation, which limits geometry preservation across a deep hierarchy. To address these gaps, we propose Graph Regularized Deep Symmetric Nonnegative Matrix Factorization (GDSNMF), a deep SNMF framework that extends single-layer symmetric factorization to multi-layer hierarchical decomposition. GDSNMF learns a composed deep representation through the product of layer-wise factors and introduces a deep similarity graph by imposing Laplacian regularization on this composed representation. Consequently, neighborhood and manifold information can be propagated through the entire hierarchy, enabling the model to jointly capture hierarchical structure and intrinsic data geometry. Moreover, GDSNMF is developed from both symmetric and nonsymmetric perspectives. The main symmetric formulation preserves the original SNMF structure and leads to a fourth-order matrix polynomial, whereas the nonsymmetric variant uses a second-order matrix polynomial as a simpler alternative. This dual formulation enables a direct investigation of the trade-off between symmetry preservation, mathematical tractability, and empirical performance. For both variants, multiplicative update rules are derived and convergence behavior is analyzed. Experiments on eleven benchmark datasets from image analysis, biology, text processing, and community detection demonstrate that the symmetric fourth-order GDSNMF generally outperforms its nonsymmetric counterpart and several classical, state-of-the-art, and recent NMF-based methods under a unified and interpretable deep factorization setting. The source code and experimental scripts are publicly available at https://github.com/saeedkarami2012/ GDSNMF.

Keywords:

Symmetric NMF

Deep symmetric NMF

Multi-layer decomposition

Graph regularization

Clustering

Graph Regularized Deep Symmetric Nonnegative Matrix Factorization

Abstract

Journal

Organization

Cited Papers