Sparse projections onto the simplex

TL;DR

This paper introduces efficient methods for projecting onto the simplex to handle non-convex constraints in high-dimensional learning problems like quantum tomography and portfolio selection.

Contribution

It develops sparse projection algorithms onto the simplex and its extension, enabling solutions to complex non-convex constrained learning problems.

Findings

Efficient algorithms for sparse projections onto the simplex.

Applications demonstrated in quantum tomography and portfolio optimization.

Improved handling of non-convex constraints in high-dimensional learning.

Abstract

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.