On regularized Radon-Nikodym differentiation
This work addresses a fundamental mathematical problem with broad applications in statistics and machine learning, though it appears incremental as it builds on existing regularization frameworks.
The paper tackles the problem of estimating Radon-Nikodym derivatives, which is important for applications like covariate shift adaptation and mutual information estimation, by using a regularization scheme in reproducing kernel Hilbert spaces and establishes convergence rates with theoretical and numerical validation.
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.