Structured Sparse Non-negative Matrix Factorization with L20-Norm for scRNA-seq Data Analysis

Non-negative matrix factorization (NMF) is a powerful tool for dimensionality reduction and clustering. Unfortunately, the interpretation of the clustering results from NMF is difficult, especially for the high-dimensional biological data without effective feature selection. In this paper, we first introduce a row-sparse NMF with $\ell_{2,0}$-norm constraint (NMF_$\ell_{20}$), where the basis matrix $W$ is constrained by the $\ell_{2,0}$-norm, such that $W$ has a row-sparsity pattern with feature selection. It is a challenge to solve the model, because the $\ell_{2,0}$-norm is non-convex and non-smooth. Fortunately, we prove that the $\ell_{2,0}$-norm satisfies the Kurdyka-\L{ojasiewicz} property. Based on the finding, we present a proximal alternating linearized minimization algorithm and its monotone accelerated version to solve the NMF_$\ell_{20}$ model. In addition, we also 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. We propose an efficient algorithm to solve ONMF_$\ell_{20}$ by transforming it into a series of constrained and penalized matrix factorization problems. The results on numerical and scRNA-seq datasets demonstrate the efficiency of our methods in comparison with existing methods.