On recovering the Radon-Nikodym derivative under the big data assumption

TL;DR

This paper introduces an algorithm combining Nyström subsampling and Tikhonov regularization to recover the Radon-Nikodym derivative efficiently under big data conditions, with proven convergence rates.

Contribution

It presents a novel method that achieves high accuracy and subquadratic computational costs for Radon-Nikodym derivative recovery in large datasets.

Findings

Convergence rates are established for derivatives in RKHS and outside RKHS.

The method attains accuracy comparable to full-sample algorithms.

Computational costs are reduced to subquadratic complexity.

Abstract

The present paper is focused on the problem of recovering the Radon-Nikodym derivative under the big data assumption. To address the above problem, we design an algorithm that is a combination of the Nystr\"om subsampling and the standard Tikhonov regularization. The convergence rate of the corresponding algorithm is established both in the case when the Radon-Nikodym derivative belongs to RKHS and in the case when it does not. We prove that the proposed approach not only ensures the order of accuracy as algorithms based on the whole sample size, but also allows to achieve subquadratic computational costs in the number of observations.