Minimax estimation of kernel mean embeddings

In this paper, we study the minimax estimation of the Bochner integral (Equation Presented) also called as the kernel mean embedding, based on random samples drawn i.i.d. from P, where k : X × X → ℝ is a positive definite kernel. Various estimators (including the empirical estimator), θ_n of µ_k(P) are studied in the literature wherein all of them satisfy ∥θ_n-µ_k(P)∥_Hk = O_P(n^-1/2) with H_k being the reproducing kernel Hilbert space induced by k. The main contribution of the paper is in showing that the above mentioned rate of n^-1/2 is minimax in ∥· ∥_Hk and ∥· ∥L2_(ℝd₎-norms over the class of discrete measures and the class of measures that has an infinitely differentiable density, with k being a continuous translation-invariant kernel on ℝ^d. The interesting aspect of this result is that the minimax rate is independent of the smoothness of the kernel and the density of P (if it exists).