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}.