Module quantile_kernel 

Kernel quantile embeddings for tail-sensitive distribution comparison. Kernel quantile embeddings for tail-sensitive distribution comparison.

Standard kernel mean embeddings map distributions to RKHS elements via mu_P = E_{x~P}[k(x, .)]. This captures the mean behavior but can miss tail differences between distributions.

Kernel quantile embeddings instead embed at each quantile level tau: the embedding at tau weights samples by their position relative to the tau-th quantile, making the comparison sensitive to distributional shape across all quantile levels.

The Quantile Maximum Mean Discrepancy (QMMD) integrates MMD over quantile levels, giving a metric that is more sensitive to tail differences than standard MMD.

Reference: Naslidnyk, Chau, Briol, Muandet (2025). “Kernel Quantile Embeddings”

Enums

QuantileWeight

Weighting scheme for quantile levels in weighted QMMD.

Functions

kernel_quantile_embedding

Kernel quantile embedding evaluated at given points.

qmmd

Quantile Maximum Mean Discrepancy between two sets of 1D samples.

quantile_distribution_kernel

Kernel between two distributions defined via their quantile embeddings.

quantile_function_embedding

Kernel quantile function embedding.

quantile_gram_matrix

Quantile kernel Gram matrix at a given quantile level.

weighted_qmmd

Weighted Quantile MMD between two sets of 1D samples.