We derive efficient sparse projections onto the simplex and its extension, and illustrate how to use them to solve high-dimensional learning problems such as portfolio selection with non-convex constraints. To this end, we study the following sparse Euclidean projections: Problem 1. (Simplex) Given w ∈ ℝp, find a Euclidean projection of w onto the intersection of k-sparse vectors Σk = {β∈ℝp : | {i : βi≠0}| ≤k} and the simplex Δλ+={β∈ℝp : βi ≥ 0, Σiβi=λ}: equation Problem 2. (Hyperplane) Replace Δλ+ in (1) with the hyperplane constraint Δλ = {β∈ℝp : Σiβi = λ}.
展开▼
机译：我们在单形及其扩展上导出有效的稀疏投影，并说明如何使用它们来解决高维学习问题，例如具有非凸约束的投资组合选择。为此，我们研究以下稀疏的欧几里得投影：问题1.（单形）给定w∈ℝ p ，在k个稀疏向量Σk= {β∈的交点上找到w的欧几里得投影。 ℝ p ：| {i：βi≠0} | ≤k}和单纯形Δλ + = {β∈ℝ p ：βi≥0，Σiβi=λ}：方程问题2。（超平面）替换Δλ （1）中的+ 具有超平面约束Δλ= {β∈ℝ p ：Σiβi=λ}。
展开▼
