Minimax estimation of kernel mean embeddings

Abstract

©2017 Ilya Tolstikhin, Bharath K. Sriperumbudur, and Krikamol Muandet. 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), θnof µk(P) are studied in the literature wherein all of them satisfy ∥θn-µk(P)∥Hk= OP(n-1/2) with Hkbeing the reproducing kernel Hilbert space induced by k. The main contribution of the paper is in showing that the above mentioned rate of n-1/2is minimax in ∥· ∥Hkand ∥· ∥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 ℝd. 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).