A Supplementary Material for “Non-Gaussian Component Analysis with Log-Density Gradient Estimation”

1 Proof of Theorem 1 Our proof can be divided into two parts as mentioned in the main manuscript. First of all, we establish the growth condition of LSLDG (see Definition 6.1 in Bonnans and Shapiro (1998) for the detailed definition of the growth condition). Denote the expected and empirical objective functions by J * j (θ) = θ ⊤ G * j θ + 2θ ⊤ h * j + λ * j θ ⊤ θ, J j (θ) = θ ⊤ G j θ + 2θ ⊤ h j + λ j θ ⊤ θ. Then θ * j = argmin θ J * j (θ) and θ j = argmin θ J j (θ), and we have Lemma 1. The following second-order growth condition holds J * j (θ) ≥ J * j (θ * j) + ϵ λ ∥θ − θ * j ∥ 2 2. Proof. J * j (θ) is strongly convex with parameter at least 2λ * j , since G * j is symmetric and positive-definite. Hence, J * j (θ) ≥ J * j (θ * j) + (∇J * j (θ * j)) ⊤ (θ − θ * j) + λ * j ∥θ − θ * j ∥ 2 2 ≥ J * j (θ * j) + ϵ λ ∥θ − θ * j ∥ 2 2 , where we used the optimality condition ∇J * j (θ * j) = 0 and the first condition λ * j ≥ ϵ λ of the theorem.