On regularized Radon-Nikodym differentiation

TL;DR

This paper introduces a regularized kernel-based method for estimating Radon-Nikodym derivatives, providing convergence rates and high-accuracy reconstruction, with theoretical analysis supported by numerical simulations.

Contribution

It develops a general regularization framework in reproducing kernel Hilbert spaces for Radon-Nikodym derivative estimation, including convergence analysis and pointwise accuracy results.

Findings

Established convergence rates considering smoothness and capacity.

Demonstrated high-order accuracy in derivative reconstruction.

Validated theoretical results with numerical simulations.

Abstract

We discuss the problem of estimating Radon-Nikodym derivatives. This problem appears in various applications, such as covariate shift adaptation, likelihood-ratio testing, mutual information estimation, and conditional probability estimation. To address the above problem, we employ the general regularization scheme in reproducing kernel Hilbert spaces. The convergence rate of the corresponding regularized algorithm is established by taking into account both the smoothness of the derivative and the capacity of the space in which it is estimated. This is done in terms of general source conditions and the regularized Christoffel functions. We also find that the reconstruction of Radon-Nikodym derivatives at any particular point can be done with high order of accuracy. Our theoretical results are illustrated by numerical simulations.