# Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions

arXiv: https://arxiv.org/abs/1604.05251 · HTML: https://mlsanity.com/paper/1604.05251/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Apr 2016.

## Built on by (most cited first, 30 of 40)

- [Conditional Distributional Treatment Effect with Kernel Conditional Mean Embeddings and U-Statistic Regression](https://mlsanity.com/paper/2102.08208/) (2021)
- [Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Survey](https://mlsanity.com/paper/2106.08443/) (2021)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [Drifting Fields are not Conservative](https://mlsanity.com/paper/2604.06333/) (2026)
- [Learning Counterfactually Invariant Predictors](https://mlsanity.com/paper/2207.09768/) (2022)
- [Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures](https://mlsanity.com/paper/2206.07588/) (2022)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [Kernel Robust Hypothesis Testing](https://mlsanity.com/paper/2203.12777/) (2022)
- [Spectral Regularized Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2308.04561/) (2023)
- [A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations](https://mlsanity.com/paper/2502.13570/) (2025)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [Estimation Beyond Data Reweighting: Kernel Method of Moments](https://mlsanity.com/paper/2305.10898/) (2023)
- [Handling Hard Affine SDP Shape Constraints in RKHSs](https://mlsanity.com/paper/2101.01519/) (2021)
- [On boundary conditions parametrized by analytic functions](https://mlsanity.com/paper/2205.03185/) (2022)
- [Robust Sample Weighting to Facilitate Individualized Treatment Rule Learning for a Target Population](https://mlsanity.com/paper/2105.00581/) (2021)
- [Reproducing kernel Hilbert spaces in the mean field limit](https://mlsanity.com/paper/2302.14446/) (2023)
- [Geometric Learning with Positively Decomposable Kernels](https://mlsanity.com/paper/2310.13821/) (2023)
- [Learning to Embed Distributions via Maximum Kernel Entropy](https://mlsanity.com/paper/2408.00549/) (2024)
- [Signature Maximum Mean Discrepancy Two-Sample Statistical Tests](https://mlsanity.com/paper/2506.01718/) (2025)
- [Non-parametric online market regime detection and regime clustering for multidimensional and path-dependent data structures](https://mlsanity.com/paper/2306.15835/) (2023)
- [Kernel-Based Testing for Single-Cell Differential Analysis](https://mlsanity.com/paper/2307.08509/) (2023)
- [Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels](https://mlsanity.com/paper/2402.08425/) (2024)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Ensemble forecasts in reproducing kernel Hilbert space family](https://mlsanity.com/paper/2207.14653/) (2022)
- [Transportation Marketplace Rate Forecast Using Signature Transform](https://mlsanity.com/paper/2401.04857/) (2024)

## Abstract (arXiv)

Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures μ from some set M to functions in a reproducing kernel Hilbert space (RKHS) with kernel k. The RKHS distance of two mapped measures is a semi-metric dk over M. We study three questions. (I) For a given kernel, what sets M can be embedded? (II) When is the embedding injective over M (in which case dk is a metric)? (III) How does the dk-induced topology compare to other topologies on M? The existing machine learning literature has addressed these questions in cases where M is (a subset of) the finite regular Borel measures. We unify, improve and generalise those results. Our approach naturally leads to continuous and possibly even injective embeddings of (Schwartz-) distributions, i.e., generalised measures, but the reader is free to focus on measures only. In particular, we systemise and extend various (partly known) equivalences between different notions of universal, characteristic and strictly positive definite kernels, and show that on an underlying locally compact Hausdorff space, dk metrises the weak convergence of probability measures if and only if k is continuous and characteristic.