aiqbench analysis
This paper introduces DKMD, a novel statistical metric designed to detect directional shifts in univariate distributions while maintaining computational efficiency through an $O(N \log N)$ algorithm. By overcoming the quadratic cost and information loss typical of squared Maximum Mean Discrepancy methods, this approach offers significant theoretical advances for distribution comparison tasks critical in model evaluation and anomaly detection.
Directional Kernel Mean Difference: A Fast Signed Statistic for Univariate Distribution Comparison
arXiv:2607.20119v1 Announce Type: cross 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.
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.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2607.20119 [stat.ML]
(or arXiv:2607.20119v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2607.20119 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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