Ensemble Estimation of Information Divergence †.

Recent work has focused on the problem of nonparametric estimation of information divergence functionals between two continuous random variables. Many existing approaches require either restrictive assumptions about the density support set or difficult calculations at the support set boundary which must be known a priori. The mean squared error (MSE) convergence rate of a leave-one-out kernel density plug-in divergence functional estimator for general bounded density support sets is derived where knowledge of the support boundary, and therefore, the boundary correction is not required. The theory of optimally weighted ensemble estimation is generalized to derive a divergence estimator that achieves the parametric rate when the densities are sufficiently smooth. Guidelines for the tuning parameter selection and the asymptotic distribution of this estimator are provided. Based on the theory, an empirical estimator of Rényi-α divergence is proposed that greatly outperforms the standard kernel density plug-in estimator in terms of mean squared error, especially in high dimensions. The estimator is shown to be robust to the choice of tuning parameters. We show extensive simulation results that verify the theoretical results of our paper. Finally, we apply the proposed estimator to estimate the bounds on the Bayes error rate of a cell classification problem.

Ensemble estimators for multivariate entropy estimation.

The problem of estimation of density functionals like entropy and mutual information has received much attention in the statistics and information theory communities. A large class of estimators of functionals of the probability density suffer from the curse of dimensionality, wherein the mean squared error (MSE) decays increasingly slowly as a function of the sample size T as the dimension d of the samples increases. In particular, the rate is often glacially slow of order O(T-γ/d ), where γ > 0 is a rate parameter. Examples of such estimators include kernel density estimators, k-nearest neighbor (k-NN) density estimators, k-NN entropy estimators, intrinsic dimension estimators and other examples. In this paper, we propose a weighted affine combination of an ensemble of such estimators, where optimal weights can be chosen such that the weighted estimator converges at a much faster dimension invariant rate of O(T-1). Furthermore, we show that these optimal weights can be determined by solving a convex optimization problem which can be performed offline and does not require training data. We illustrate the superior performance of our weighted estimator for two important applications: (i) estimating the Panter-Dite distortion-rate factor and (ii) estimating the Shannon entropy for testing the probability distribution of a random sample.

GLOBAL PERFORMANCE PREDICTION FOR DIVERGENCE-BASED IMAGE REGISTRATION CRITERIA.

Divergence measures find application in many areas of statistics, signal processing and machine learning, thus necessitating the need for good estimators of divergence measures. While several estimators of divergence measures have been proposed in literature, the performance of these estimators is not known. We propose a simple kNN density estimation based plug-in estimator for estimation of divergence measures. Based on the properties of kNN density estimates, we derive the bias, variance and mean square error xof the estimator in terms of the sample size, the dimension of the samples and the underlying probability distribution. Based on these results, we specify the optimal choice of tuning parameters for minimum mean square error. We also present results on convergence in distribution of the proposed estimator. These results will establish a basis for analyzing the performance of image registration methods that maximize divergence.