  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.
