🤖 AI Summary
This work addresses the inefficiency of symmetric nonnegative matrix factorization (Symmetric NMF) in graph clustering and related tasks, which stems from the slow convergence of conventional projected gradient methods. To overcome this limitation, the authors propose SNMPBB, the first algorithm to integrate a nonmonotone projected Barzilai–Borwein method into Symmetric NMF, achieving both rapid and globally convergent optimization while preserving curvature information for enhanced efficiency. The framework is further extended to Graph-SNMPBB, incorporating graph Laplacian regularization for improved clustering performance, and LAI-SNMPBB, which combines low-rank approximation with randomization for scalability to large sparse matrices. Experiments demonstrate that SNMPBB is six times faster than SymANLS on synthetic data; Graph-SNMPBB matches or exceeds state-of-the-art accuracy on six real-world clustering benchmarks; and LAI-SNMPBB significantly outperforms LAI-SymPGNCG across 34 SuiteSparse matrices.
📝 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.