  This paper deals with unsupervised clustering with feature selection. The
problem is to estimate both labels and a sparse projection matrix of weights.
To address this combinatorial non-convex problem maintaining a strict control
on the sparsity of the matrix of weights, we propose an alternating
minimization of the Frobenius norm criterion. We provide a new efficient
algorithm named K-sparse which alternates k-means with projection-gradient
minimization. The projection-gradient step is a method of splitting type, with
exact projection on the $\ell^1$ ball to promote sparsity. The convergence of
the gradient-projection step is addressed, and a preliminary analysis of the
alternating minimization is made. The Frobenius norm criterion converges as the
number of iterates in Algorithm K-sparse goes to infinity. Experiments on
Single Cell RNA sequencing datasets show that our method significantly improves
the results of PCA k-means, spectral clustering, SIMLR, and Sparcl methods, and
achieves a relevant selection of genes. The complexity of K-sparse is linear in
the number of samples (cells), so that the method scales up to large datasets.
