# On regularized Radon-Nikodym differentiation

arXiv: https://arxiv.org/abs/2308.07887 · HTML: https://mlsanity.com/paper/2308.07887/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 25 references that are in the map.
- First posted on arXiv 15 Aug 2023.

## Builds on (most cited first, 5 of 5)

- [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015)
- [Kernel Conditional Density Operators](https://mlsanity.com/paper/1905.11255/) (2019)
- [Relating Leverage Scores and Density using Regularized Christoffel Functions](https://mlsanity.com/paper/1805.07943/) (2018)
- [Inverse Density as an Inverse Problem: The Fredholm Equation Approach](https://mlsanity.com/paper/1304.5575/) (2013)
- [Analysis of regularized Nystr"om subsampling for regression functions of low smoothness](https://mlsanity.com/paper/1806.00826/) (2018)

## Built on by (most cited first, 10 of 10)

- [Overcoming Saturation in Density Ratio Estimation by Iterated Regularization](https://mlsanity.com/paper/2402.13891/) (2024)
- [Projection Pursuit Density Ratio Estimation](https://mlsanity.com/paper/2506.00866/) (2025)
- [Diffusion Secant Alignment for Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2509.04852/) (2025)
- [Spectral Algorithms in Misspecified Regression: Convergence under Covariate Shift](https://mlsanity.com/paper/2509.05106/) (2025)
- [Binary Losses for Density Ratio Estimation](https://mlsanity.com/paper/2407.01371/) (2024)
- [Kernel Density Machines](https://mlsanity.com/paper/2504.21419/) (2025)
- [Towards regularized learning from functional data with covariate shift](https://mlsanity.com/paper/2601.21019/) (2026)
- [One-Step Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2604.10672/) (2026)
- [Learning density ratios in causal inference using Bregman-Riesz regression](https://mlsanity.com/paper/2510.16127/) (2025)
- [Convergence Analysis of Nystr"om Subsampling in Covariate Shift Adaptation for Misspecified case](https://mlsanity.com/paper/2606.22259/) (2026)

## Abstract (arXiv)

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.