Kernel Quantile Embeddings and Associated Probability Metrics

TL;DR

This paper introduces kernel quantile embeddings (KQEs) to create new probability metrics that are computationally efficient, weaker than MMD, and include a kernelized sliced Wasserstein distance, with competitive hypothesis testing performance.

Contribution

The paper proposes kernel quantile embeddings (KQEs) as a novel RKHS representation that leads to new probability metrics with weaker kernel conditions and efficient estimation methods.

Findings

KQEs define new probability metrics weaker than MMD.

The metrics include a kernelized sliced Wasserstein distance.

Empirical tests show competitive performance in hypothesis testing.

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.