The Nearest Neighbor Information Estimator is Adaptively Near Minimax Rate-Optimal

Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)

Bibtex Metadata Paper Reviews Supplemental

Authors

Jiantao Jiao, Weihao Gao, Yanjun Han

Abstract

We analyze the Kozachenko–Leonenko (KL) fixed k-nearest neighbor estimator for the differential entropy. We obtain the first uniform upper bound on its performance for any fixed k over H\"{o}lder balls on a torus without assuming any conditions on how close the density could be from zero. Accompanying a recent minimax lower bound over the H\"{o}lder ball, we show that the KL estimator for any fixed k is achieving the minimax rates up to logarithmic factors without cognizance of the smoothness parameter s of the H\"{o}lder ball for $s \in (0,2]$ and arbitrary dimension d, rendering it the first estimator that provably satisfies this property.