Kernel Quantile Embeddings and Associated Probability Metrics

Masha Naslidnyk, Siu Lun Chau, Francois-Xavier Briol, Krikamol Muandet
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:45770-45800, 2025.

Abstract

Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that: (i) are probability metrics under weaker kernel conditions than MMD; (ii) recover a kernelised form of the sliced Wasserstein distance; and (iii) can be efficiently estimated with near-linear cost. Through hypothesis testing, we show that these distances offer a competitive alternative to MMD and its fast approximations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-naslidnyk25a, title = {Kernel Quantile Embeddings and Associated Probability Metrics}, author = {Naslidnyk, Masha and Chau, Siu Lun and Briol, Francois-Xavier and Muandet, Krikamol}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {45770--45800}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/naslidnyk25a/naslidnyk25a.pdf}, url = {https://proceedings.mlr.press/v267/naslidnyk25a.html}, abstract = {Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that: (i) are probability metrics under weaker kernel conditions than MMD; (ii) recover a kernelised form of the sliced Wasserstein distance; and (iii) can be efficiently estimated with near-linear cost. Through hypothesis testing, we show that these distances offer a competitive alternative to MMD and its fast approximations.} }
Endnote
%0 Conference Paper %T Kernel Quantile Embeddings and Associated Probability Metrics %A Masha Naslidnyk %A Siu Lun Chau %A Francois-Xavier Briol %A Krikamol Muandet %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-naslidnyk25a %I PMLR %P 45770--45800 %U https://proceedings.mlr.press/v267/naslidnyk25a.html %V 267 %X Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that: (i) are probability metrics under weaker kernel conditions than MMD; (ii) recover a kernelised form of the sliced Wasserstein distance; and (iii) can be efficiently estimated with near-linear cost. Through hypothesis testing, we show that these distances offer a competitive alternative to MMD and its fast approximations.
APA
Naslidnyk, M., Chau, S.L., Briol, F. & Muandet, K.. (2025). Kernel Quantile Embeddings and Associated Probability Metrics. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:45770-45800 Available from https://proceedings.mlr.press/v267/naslidnyk25a.html.

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