Supplementary material : Sparsity and the truncated ` 2 - norm

We require a basic lemma on soft-threhsolding before proceeding with the proof of Theorem 1. Recall the soft-thresholding function s λ (x) = sign(x){(|x| − λ) ∨ 0} (x ∈ R, λ ≥ 0) and define r(λ; θ) = E θ {s λ (x) − θ} 2 , θ ∈ R, λ ≥ 0 to be the risk of soft-thresholding in the 1-dimensional problem, where x ∼ N (θ, 1). The following result is essentially contained in (Johnstone, 2013).