An alternating minimization algorithm for sparse convolutive non-negative matrix factorization with ${\ell }_1$-norm

Convolutive non-negative matrix factorization has been a dominant analytical technique for deriving interpretable insights from data in speech processing, image analysis, data mining, biomedicine, and other fields. In this paper, a sparse convolutive non-negative matrix factorization model was introduced by incorporating an ${\ell }_1$ regularization on representation matrices. This enhancement not only preserved the inherent characteristics of convolutive non-negative matrix factorization, but also promoted sparse data representation, thereby facilitating more efficient data storage and analysis. An alternating minimization algorithm for the presented model was proposed by integrating the alternating direction method of multipliers with the accelerated iterative shrinkage-thresholding algorithm. In addition, a convergence result was presented that the convergence point of the algorithm necessarily constitutes a stable point of the problem. Experimental results showed that the proposed algorithm yielded sparser solutions for synthetic data designed to simulate sparse representation scenarios, and achieved practical applicability in speech dataset, validating its potential for real-world signal processing tasks.