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 Hölder 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 Hölder 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 Hölder ball for s (0, 2] and arbitrary dimension d, rendering it the first estimator that provably satisfies this property.

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 precomputed 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.

Estimation of mutual information from observed samples is a basic primitive in machine learning, useful in several learning tasks including correlation mining, information bottleneck, Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a quantity well-defined for general probability spaces, estimators have been developed only in the special case of discrete or continuous pairs of random variables. Most of these estimators operate using the 3H-principle, i.e., by calculating the three (differential) entropies of X, Y and the pair (X, Y). However, in general mixture spaces, such individual entropies are not well defined, even though mutual information is. In this paper, we develop a novel estimator for estimating mutual information in discrete-continuous mixtures. We prove the consistency of this estimator theoretically as well as demonstrate its excellent empirical performance. This problem is relevant in a wide-array of applications, where some variables are discrete, some continuous, and others are a mixture between continuous and discrete components.

Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.

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 Hölder 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 Hölder 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 Hölder ball for s (0, 2] and arbitrary dimension d, rendering it the first estimator that provably satisfies this property.

Neural Information Processing Systems http://nips.cc/

Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.