用核函数定义新分布表示法，提升概率距离计算的效率与适用性
Kernel Quantile Embeddings and Associated Probability Metrics
- 提出核量化嵌入，用核方法表示分布的分位数特征
- 新距离在更弱核条件下仍为有效概率度量，且可高效近似计算
- 适合需要快速、稳定分布比较的研究场景
将概率分布嵌入再生核希尔伯特空间（RKHS）已催生强大非参数方法，如最大均值差异（MMD），其依赖核均值嵌入将分布表示为RKHS中的均值函数。然而，均值函数是否为唯一有意义的表示尚不明确。受广义分位数启发，本文引入核量化嵌入（KQE），并基于此构建一类距离：（i）在比MMD更弱的核条件下仍为概率度量；（ii）可恢复核化切片沃尔什距离形式；（iii）支持近线性复杂度高效估计。通过假设检验验证，该距离在性能上可媲美MMD及其快速近似方法。
原文摘要 · Abstract (English)
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.
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