发表机构
Northwestern Polytechnical University（西北工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出方向核均值差异（DKMD）用于单变量分布比较，通过积分核均值嵌入差异与奇权重函数保留分布变化方向，推导数据驱动估计器，开发高效算法，实验证明其能准确分离方向变化、抗重尾异常值且计算高效。
AI 中文摘要
我们引入了方向核均值差异（DKMD），这是一种用于单变量分布比较的有符号统计量，可保留分布变化的方向。与通过对RKHS距离平方而丢弃方向信息的平方最大均值差异（MMD）不同，DKMD将核均值嵌入的差异与固定的奇权重函数进行积分。这种构造产生了三个结构特性：反对称性、对对称分布差异的免疫性以及在随机占优下的方向单调性。我们推导了一个数据驱动的黎曼估计器，确保与连续公式渐近一致，在实证评估中严格保留有符号统计量的理论保证。为克服核方法的二次计算成本，我们开发了一种$O(N \log N)$的前缀 - 后缀扫描算法，该算法利用实数线的全序，同时仅需$O(N)$内存。在合成基准上的实验表明，DKMD能从对称扰动中正确分离方向变化，对可能翻转均值差异符号的重尾异常值保持稳健，并能在数秒内扩展到数百万样本。
英文摘要
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.
