Recursive partitioning methods provide computationally efficient surrogates for the Wasserstein distance, yet their statistical behavior and their resolution in the small-discrepancy regime remain insufficiently understood. We study Recursive Rank Matching (RRM) as a representative instance of this class under a population-anchored reference. In this setting, we establish consistency and an explicit convergence rate for the anchored empirical RRM under the quadratic cost. We then identify a dominant mismatch mechanism responsible for the loss of resolution in the small-discrepancy regime. Based on this analysis, we introduce Selective Recursive Rank Matching (SRRM), which suppresses the resulting dominant mismatches and yields a higher-fidelity practical surrogate for the Wasserstein distance at moderate additional computational cost.

翻译：递归划分方法为Wasserstein距离提供了计算高效的代理，但其统计行为及其在小差异区域内的分辨率尚未得到充分理解。我们以递归秩匹配（RRM）为这类方法的代表实例，在基于总体锚定的参考框架下展开研究。在此设定中，我们建立了二次代价下锚定经验RRM的一致性及显式收敛速率。随后，我们识别出导致小差异区域分辨率损失的主导不匹配机制。基于此分析，我们提出选择性递归秩匹配（SRRM），该方法抑制了由此产生的主导不匹配，并以适度增加的计算成本获得了Wasserstein距离的高保真实用代理。