Breaking the Bandwidth Barrier: Geometrical Adaptive Entropy Estimation

TL;DR

This paper introduces a novel entropy and mutual information estimator combining geometric and kernel methods with local bandwidth selection, achieving improved accuracy and a universal bias correction.

Contribution

It proposes a new estimator that integrates geometric and kernel approaches with local bandwidths, and demonstrates its universal bias properties and theoretical foundations.

Findings

Outperforms existing state-of-the-art estimators

Bias can be pre-computed and corrected universally

Provides a unified framework for kernel and NN estimators

Abstract

Estimators of information theoretic measures such as entropy and mutual information are a basic workhorse for many downstream applications in modern data science. State of the art approaches have been either geometric (nearest neighbor (NN) based) or kernel based (with a globally chosen bandwidth). In this paper, we combine both these approaches to design new estimators of entropy and mutual information that outperform state of the art methods. Our estimator uses local bandwidth choices of $k$-NN distances with a finite $k$, independent of the sample size. Such a local and data dependent choice improves performance in practice, but the bandwidth is vanishing at a fast rate, leading to a non-vanishing bias. We show that the asymptotic bias of the proposed estimator is universal; it is independent of the underlying distribution. Hence, it can be pre-computed and subtracted from the estimate. As a byproduct, we obtain a unified way of obtaining both kernel and NN estimators. The corresponding theoretical contribution relating the asymptotic geometry of nearest neighbors to order statistics is of independent mathematical interest.