Separable nonnegative matrix factorization (SNMF) has been widely used for low-rank representation and clustering of nonnegative data, owing to its ability to produce part-based and interpretable decompositions. In particular, SNMF is closely related to graph clustering and community detection. To enhance sparsity and identifiability of the learned factors, we propose an $\ell_1^p/\ell_2$-regularized SNMF model based on a powered ratio-of-norms regularizer. The resulting formulation is nonconvex and nonsmooth, which poses significant challenges for optimization. To address this, we develop efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM). The proposed methods decompose the original problem into tractable subproblems, leveraging closed-form proximal operators associated with the powered norm terms. We establish descent and limiting criticality properties for the DCA scheme and convergence under standard assumptions for the ADMM scheme. Extensive numerical experiments on synthetic datasets and hand gesture classification tasks demonstrate that the proposed approach achieves competitive or improved performance in anchor identification and classification accuracy compared with existing SNMF methods, while maintaining competitive computational efficiency.
Qianqian Qi, Zhongming Chen, Peter G. M. van der Heijdencs.LG
Nonnegative matrix factorization (NMF) decomposes a nonnegative matrix into the product of two nonnegative matrices. This property makes NMF well suited for unmixing grain-size distribution data, which are inherently nonnegative and have row sums equal to one. Previous studies have shown that AnalySize, an NMF-based method, performs well on poorly mixed grain-size distribution data but struggles when the data is highly mixed, where no observed samples are close to the true end members. To overcome this limitation, we introduce a maximum-distance NMF that encourages the estimated end members to be as distinct as possible and develop a hierarchical alternating least squares algorithm for optimization. The proposed formulation can be regarded as a generalization of AnalySize, where AnalySize minimizes the distance among end members while the proposed method maximizes it. Experimental results demonstrate that the method effectively decomposes highly mixed grain-size distribution data.
Nonnegative Matrix Factorization (NMF) is a fundamental tool in unsupervised learning, which approximates a nonnegative matrix by the product of two low-rank nonnegative factors. The Kullback-Leibler (KL) divergence is best suited to measure the data to model discrepancy when the decomposed data sample follows a Poisson distribution, which is the case for count datasets such as term-document matrices or images. Most KL-NMF algorithms in the literature minimize a separable majorant of the loss to find their next iterate. We argue that this method has reached its limits and propose to use instead the second-order Taylor expansion of the loss, leading to a Newton-type method. We minimize this non-separable surrogate by proposing a generalization of the well-known HALS algorithm. This yields an efficient KL-NMF algorithm which provably converges and which competes favorably with state-of-the-art algorithms on a large variety of datasets.
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.