Sparse projections onto the simplex
This addresses optimization challenges in machine learning for applications like quantum tomography and portfolio selection, though it appears incremental as it builds on existing projection methods.
The paper tackled the problem of learning with rank or sparsity constraints where convex norms are also present as constraints, by deriving efficient sparse projections onto the simplex and its extension, and applied these to solve high-dimensional problems in quantum tomography, sparse density estimation, and portfolio selection.
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.