Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the ℓ1-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.
@article{arxiv.1206.1529,
title = {Sparse projections onto the simplex},
author = {Anastasios Kyrillidis and Stephen Becker and Volkan Cevher and and Christoph Koch},
journal= {arXiv preprint arXiv:1206.1529},
year = {2013}
}