Demystifying Fixed k-Nearest Neighbor Information Estimators

Abstract

Estimating mutual information from i.i.d. samples drawn from an unknown joint density function is a basic statistical problem of broad interest with multitudinous applications. The most popular estimator is one proposed by Kraskov and Stögbauer and Grassberger (KSG) in 2004, and is nonparametric and based on the distances of each sample to its $k^{\mathrm{th}}$ nearest neighboring sample, where $k$ is a fixed small integer. Despite its widespread use (part of scientific software packages), theoretical properties of this estimator have been largely unexplored. In this paper we demonstrate that the estimator is consistent and also identify an upper bound on the rate of convergence of the bias as a function of number of samples. We argue that the superior performance benefits of the KSG estimator stems from a curious "correlation boosting" effect and build on this intuition to modify the KSG estimator in novel ways to construct a superior estimator. As a byproduct of our investigations, we obtain nearly tight rates of convergence of the ${\ell }_2$ error of the well known fixed $k$ nearest neighbor estimator of differential entropy by Kozachenko and Leonenko.

55 pages, 8 figures

References in corpus (8)

• Estimation of Rényi Entropy and Mutual Information Based on Generalized Nearest-Neighbor Graphs
• Efficient Estimation of Mutual Information for Strongly Dependent Variables
• Discovering Structure in High-Dimensional Data Through Correlation Explanation
• A non-parametric k-nearest neighbour entropy estimator
• Ensemble estimation of multivariate f-divergence
• Pruning nearest neighbor cluster trees
• Exponential Concentration of a Density Functional Estimator
• Estimating Mutual Information by Local Gaussian Approximation

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