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

Researchers have introduced two new algorithms, SNMPBB and GNRBMF, for Symmetric Nonnegative Matrix Factorization (Symmetric NMF) and color image recognition, achieving significant improvements in speedup and accuracy.

The SNMPBB algorithm, introduced in a paper submitted to arXiv on June 1, 2026^[1], is the first adaptation of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF. It achieves 6 times speedup over SymANLS for similar residuals on synthetic data. Graph-SNMPBB, an extension of SNMPBB, matches or exceeds SymANLS accuracy on six real-world clustering benchmarks. Another paper, also submitted to arXiv in June 2026^[2], proposes GNRBMF, a graph regularized non-negative reduced biquaternion matrix factorization model for color image recognition. GNRBMF incorporates a graph Laplacian regularizer to improve discriminative ability and retains the non-negativity-preserving property of NRBMF. A component-wise alternating projected gradient algorithm is derived to solve the optimization problem. LAI-SNMPBB, a variant of SNMPBB, outperforms state-of-the-art LAI-SymPGNCG on 34 SuiteSparse matrices in both runtime and residual quality^[1].