SRRM: Improving Recursive Transport Surrogates in the Small-Discrepancy Regime

TL;DR

This paper introduces SRRM, a novel recursive transport surrogate method that enhances resolution in the small-discrepancy regime by suppressing dominant mismatches, supported by theoretical analysis and empirical validation.

Contribution

It proposes SRRM, an improved recursive transport surrogate method that addresses resolution issues in the small-discrepancy regime, with theoretical guarantees and practical benefits.

Findings

SRRM achieves higher fidelity in small-discrepancy regimes.

Theoretical convergence rates are established for RRM.

SRRM suppresses dominant mismatches, improving surrogate accuracy.

Abstract

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.