Title: A Nonmonotone Gradient-Based Algorithm for Symmetric Nonnegative Matrix Factorization and Graph Clustering

Original: arXiv:2606.02887v1 Announce Type: new Abstract: Symmetric nonnegative matrix factorization (Symmetric NMF) approximates a matrix as $WW^T$ with nonnegative rectangular factor $W$. It has broad applications in graph clustering and machine learning. In contrast to the NMF, projected gradient methods for the symmetric problem had been associated with slow convergence. To address this, we introduce SNMPBB, the first adaptation of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF, demonstrating that gradient algorithms are significantly more effective than previously understood. We further extend SNMPBB to graph clustering using the graph Laplacian regularization (Graph-SNMPBB) and to large problems with low-rank approximations (LAI-SNMPBB). For all variants we prove global convergence to first-order stationary points and also that Barzilai-Borwein curvature information is preserved with randomized approximations. On synthetic data, SNMPBB achieves 6 times speedup over the alternative SymANLS for similar residuals, with advantages growing at higher ranks. Across six real-world clustering benchmarks, Graph-SNMPBB matches or exceeds SymANLS accuracy. Lastly, LAI-SNMPBB outperforms state-of-the-art LAI-SymPGNCG on 34 SuiteSparse matrices in both runtime and residual quality.

Rewritten:

Title: Nonmonotone Gradient Techniques for Symmetric Nonnegative Matrix Factorization and Graph Clustering

Symmetric nonnegative matrix factorization (Symmetric NMF) serves to approximate a matrix using the form $WW^T$, where $W$ is a rectangular matrix constrained to nonnegative values. This technique is widely utilized in machine learning and graph clustering. Unlike standard NMF, projected gradient approaches applied to the symmetric variant have historically suffered from sluggish convergence rates. To overcome this limitation, we present SNMPBB, which represents the inaugural application of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF. Our findings indicate that gradient-based algorithms possess greater efficacy than previously recognized.

We subsequently develop two extensions of SNMPBB: Graph-SNMPBB, which incorporates graph Laplacian regularization for graph clustering tasks, and LAI-SNMPBB, designed to handle large-scale problems through low-rank approximations. We establish that all these variants converge globally to first-order stationary points. Additionally, we demonstrate that randomized approximations retain the Barzilai-Borwein curvature information.

Empirical evaluations on synthetic datasets reveal that SNMPBB operates six times faster than the competing SymANLS method while maintaining comparable residual levels; this performance gap widens as matrix ranks increase. In tests involving six real-world clustering benchmarks, Graph-SNMPBB achieved accuracy levels that either equaled or surpassed those of SymANLS. Finally, regarding large-scale instances, LAI-SNMPBB demonstrated superior performance compared to the current best method, LAI-SymPGNCG, across 34 SuiteSparse matrices, showing improvements in both execution time and residual precision.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC