Directional Kernel Mean Difference: A Fast Signed Statistic for Univariate Distribution Comparison

2026-07-22

AI Summary

This work addresses a critical limitation of the classical squared Maximum Mean Discrepancy (MMD), which discards directional information about distributional shifts and thus cannot distinguish symmetric perturbations from genuine unidirectional changes. To overcome this, the authors propose the Directional Kernel Mean Discrepancy (DKMD), a signed statistic that preserves shift directionality by incorporating odd-function weighting in a reproducing kernel Hilbert space. DKMD is the first kernel-based method to simultaneously achieve direction sensitivity, immunity to symmetric perturbations, and monotonicity under stochastic dominance, while rigorously preserving these theoretical properties in empirical estimation. Coupled with a Riemann data-driven estimator and an O(N log N) prefix-suffix scanning algorithm, DKMD accurately disentangles directional shifts from symmetric noise on synthetic data, exhibits robustness to heavy-tailed outliers, and scales efficiently to million-sample datasets within seconds—significantly outperforming existing approaches.

Abstract

We introduce the Directional Kernel Mean Difference (DKMD), a signed statistic for univariate distribution comparison that preserves the direction of distributional shifts. Unlike the squared Maximum Mean Discrepancy (MMD), which discards directional information by squaring the RKHS distance, DKMD integrates the difference of kernel mean embeddings against a fixed odd weighting function. This construction yields three structural properties: antisymmetry, immunity to symmetric distributional differences, and directional monotonicity under stochastic dominance. We derive a data-driven Riemann estimator that ensures asymptotic consistency with the continuous formulation, strictly preserving the theoretical guarantees of the signed statistic in empirical evaluations. To overcome the quadratic computational cost of kernel methods, we develop an $O(N \log N)$ prefix--suffix scanning algorithm that exploits the total order of the real line while requiring only $O(N)$ memory. Experiments on synthetic benchmarks demonstrate that DKMD correctly isolates directional shifts from symmetric perturbations, remains robust to heavy-tailed outliers that can flip the sign of the mean difference, and scales to millions of samples in seconds.

Problem

• distribution comparison
• directional shift
• kernel mean embedding
• signed statistic
• univariate distribution

Innovation

• Directional Kernel Mean Difference
• signed statistic
• stochastic dominance
• O(N log N) algorithm
• kernel mean embedding