Non-Asymptotic Uniform Rates of Consistency for k-NN Regression

We derive high-probability finite-sample uniform rates of consistency for $k$-NN regression that are optimal up to logarithmic factors under mild assumptions. We moreover show that $k$-NN regression adapts to an unknown lower intrinsic dimension automatically. We then apply the $k$-NN regression rates to establish new results about estimating the level sets and global maxima of a function from noisy observations.Comment: In Proceedings of 33rd AAAI Conference on Artificial Intelligence (AAAI 2019

The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables

We propose a new algorithm to compute numerically the distribution function of the sum of $d$ dependent, non-negative random variables with given joint distribution.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ284 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Renormalization and asymptotic expansion of Dirac's polarized vacuum

We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no `real' electron. We show that it admits an asymptotic expansion to any order in powers of the physical coupling constant \alphaph, provided that the ultraviolet cut-off behaves as \Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1. The renormalization parameter $

Ensemble estimation of multivariate f-divergence

f-divergence estimation is an important problem in the fields of information theory, machine learning, and statistics. While several divergence estimators exist, relatively few of their convergence rates are known. We derive the MSE convergence rate for a density plug-in estimator of f-divergence. Then by applying the theory of optimally weighted ensemble estimation, we derive a divergence estimator with a convergence rate of O(1/T) that is simple to implement and performs well in high dimensions. We validate our theoretical results with experiments.Comment: 14 pages, 6 figures, a condensed version of this paper was accepted to ISIT 2014, Version 2: Moved the proofs of the theorems from the main body to appendices at the en