Sparse projections onto the simplex

Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the ${\mathrm{\ell }}_1$-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.