  Based on a new atomic norm, we propose a new convex formulation for sparse
matrix factorization problems in which the number of nonzero elements of the
factors is assumed fixed and known. The formulation counts sparse PCA with
multiple factors, subspace clustering and low-rank sparse bilinear regression
as potential applications. We compute slow rates and an upper bound on the
statistical dimension of the suggested norm for rank 1 matrices, showing that
its statistical dimension is an order of magnitude smaller than the usual
$\ell\_1$-norm, trace norm and their combinations. Even though our convex
formulation is in theory hard and does not lead to provably polynomial time
algorithmic schemes, we propose an active set algorithm leveraging the
structure of the convex problem to solve it and show promising numerical
results.
