Structured Sparse Non-Negative Matrix Factorization With $\ell _{2,0}$ℓ2,0-Norm

Wenwen Min, Taosheng Xu, Xiang Wan, Tsung-Hui Chang

Abstract: Non-negative matrix factorization (NMF) is a powerful tool for dimensionality reduction and clustering. However, the interpretation of the clustering result from NMF is difficult, especially for the high-dimensional biological data without effective feature selection. To address this problem, we introduce a row-sparse NMF with $\ell _{2,0}$-norm constraint (NMF_$\ell _{20}$), where the basis matrix $\boldsymbol{W}$ is constrained by using the $\ell _{2,0}$-norm constraint such that $\boldsymbol{W}$ has a row-sparsity pattern with feature selection. However, it is a challenge to solve the model, because the $\ell _{2,0}$-norm constraint is a non-convex and non-smooth function. Fortunately, we prove that the $\ell _{2,0}$-norm constraint satisfies the Kurdyka-Łojasiewicz property. Based on this finding, we present a proximal alternating linearized minimization algorithm and its monotone accelerated version to solve the NMF_$\ell _{20}$ model. In addition, we further present a orthogonal NMF with $\ell _{2,0}$-norm constraint (ONMF_$\ell _{20}$) to enhance the clustering performance by using a non-negative orthogonal constraint. The ONMF_$\ell _{20}$ model is solved by transforming into a series of constrained and penalized matrix factorization problems. The convergence and guarantees for these proposed algorithms are proved and the computational complexity is well evaluated. The results on numerical and scRNA-seq datasets demonstrate the efficiency of our methods in comparison with existing methods.