  Most learning methods with rank or sparsity constraints use convex
relaxations, which lead to optimization with the nuclear norm or the
$\ell_1$-norm. However, several important learning applications cannot benefit
from this approach as they feature these convex norms as constraints in
addition to the non-convex rank and sparsity constraints. In this setting, we
derive efficient sparse projections onto the simplex and its extension, and
illustrate how to use them to solve high-dimensional learning problems in
quantum tomography, sparse density estimation and portfolio selection with
non-convex constraints.
