# Measure transport with kernel mean embeddings

arXiv: https://arxiv.org/abs/2401.12967 · HTML: https://mlsanity.com/paper/2401.12967/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2024.

## Built on by (most cited first, 11 of 11)

- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)
- [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024)
- [Path-Guided Particle-based Sampling](https://mlsanity.com/paper/2412.03312/) (2024)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [Connections between sequential Bayesian inference and evolutionary dynamics](https://mlsanity.com/paper/2411.16366/) (2024)
- [Score-based deterministic density sampling](https://mlsanity.com/paper/2504.18130/) (2025)
- [An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2511.18060/) (2025)
- [To discretize continually: Mean shift interacting particle systems for Bayesian inference](https://mlsanity.com/paper/2605.14142/) (2026)
- [Learning Paths for Dynamic Measure Transport: A Control Perspective](https://mlsanity.com/paper/2511.03797/) (2025)
- [Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2609.18118/) (2026)

## Abstract (arXiv)

Kalman filters constitute a scalable and robust methodology for approximate Bayesian inference, matching first and second order moments of the target posterior. To improve the accuracy in nonlinear and non-Gaussian settings, we extend this principle to include more or different characteristics, based on kernel mean embeddings (KMEs) of probability measures into reproducing kernel Hilbert spaces. Focusing on the continuous-time setting, we develop a family of interacting particle systems (termed textitKME-dynamics) that bridge between prior and posterior, and that include the Kalman-Bucy filter as a special case. KME-dynamics does not require the score of the target, but rather estimates the score implicitly and intrinsically, and we develop links to score-based generative modeling and importance reweighting. A variant of KME-dynamics has recently been derived from an optimal transport and Fisher-Rao gradient flow perspective by Maurais and Marzouk, and we expose further connections to (kernelised) diffusion maps, leading to a variational formulation of regression type. Finally, we conduct numerical experiments on toy examples and the Lorenz 63 and 96 models, comparing our results against the ensemble Kalman filter and the mapping particle filter (Pulido and van Leeuwen, 2019, J. Comput. Phys.). Our experiments show particular promise for a hybrid modification (called Kalman-adjusted KME-dynamics).