Kernel Quantile Embeddings and Associated Probability Metrics
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910969638158336 |
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| author | Naslidnyk, Masha Chau, Siu Lun Briol, François-Xavier Muandet, Krikamol |
| author_facet | Naslidnyk, Masha Chau, Siu Lun Briol, François-Xavier Muandet, Krikamol |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kernel Quantile Embeddings and Associated Probability Metrics Naslidnyk, Masha Chau, Siu Lun Briol, François-Xavier Muandet, Krikamol Machine Learning Statistics Theory 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. |
| title | Kernel Quantile Embeddings and Associated Probability Metrics |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2505.20433 |