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

arXiv: https://arxiv.org/abs/2607.20119 · HTML: https://mlsanity.com/paper/2607.20119/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 12 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Builds on (most cited first, 3 of 3)

- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [Interpretable Distribution Features with Maximum Testing Power](https://mlsanity.com/paper/1605.06796/) (2016)
- [Fast Two-Sample Testing with Analytic Representations of Probability Measures](https://mlsanity.com/paper/1506.04725/) (2015)

## Abstract (arXiv)

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.