# Kernel Conditional Density Operators

arXiv: https://arxiv.org/abs/1905.11255 · HTML: https://mlsanity.com/paper/1905.11255/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2019.

## Built on by (most cited first, 14 of 14)

- [RKHS-SHAP: Shapley Values for Kernel Methods](https://mlsanity.com/paper/2110.09167/) (2021)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [On regularized Radon-Nikodym differentiation](https://mlsanity.com/paper/2308.07887/) (2023)
- [Adaptive learning of density ratios in RKHS](https://mlsanity.com/paper/2307.16164/) (2023)
- [Overcoming Saturation in Density Ratio Estimation by Iterated Regularization](https://mlsanity.com/paper/2402.13891/) (2024)
- [One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators](https://mlsanity.com/paper/2605.06873/) (2026)
- [Adaptive joint distribution learning](https://mlsanity.com/paper/2110.04829/) (2021)
- [Forward Operator Estimation in Generative Models with Kernel Transfer Operators](https://mlsanity.com/paper/2112.00305/) (2021)
- [General regularization in covariate shift adaptation](https://mlsanity.com/paper/2307.11503/) (2023)
- [Binary Losses for Density Ratio Estimation](https://mlsanity.com/paper/2407.01371/) (2024)
- [Kernel Density Machines](https://mlsanity.com/paper/2504.21419/) (2025)
- [Sequence Modeling with Spectral Mean Flows](https://mlsanity.com/paper/2510.15366/) (2025)
- [Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo](https://mlsanity.com/paper/2410.18993/) (2024)
- [Nonparametric Distribution Regression Re-calibration](https://mlsanity.com/paper/2602.13362/) (2026)

## Abstract (arXiv)

We introduce a novel conditional density estimation model termed the conditional density operator (CDO). It naturally captures multivariate, multimodal output densities and shows performance that is competitive with recent neural conditional density models and Gaussian processes. The proposed model is based on a novel approach to the reconstruction of probability densities from their kernel mean embeddings by drawing connections to estimation of Radon-Nikodym derivatives in the reproducing kernel Hilbert space (RKHS). We prove finite sample bounds for the estimation error in a standard density reconstruction scenario, independent of problem dimensionality. Interestingly, when a kernel is used that is also a probability density, the CDO allows us to both evaluate and sample the output density efficiently. We demonstrate the versatility and performance of the proposed model on both synthetic and real-world data.