SRRM: Improving Recursive Transport Surrogates in the Small-Discrepancy Regime
This work addresses a specific bottleneck in computational optimal transport for researchers and practitioners, offering an incremental improvement over existing recursive methods.
The paper tackled the problem of insufficient statistical understanding and resolution loss in recursive partitioning surrogates for the Wasserstein distance in the small-discrepancy regime, resulting in the introduction of Selective Recursive Rank Matching (SRRM) that yields a higher-fidelity surrogate at moderate additional computational cost.
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.