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

Yufei Zhang, Tao Wang, Jingyi Zhang

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.

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

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.
Paper Structure (29 sections, 7 theorems, 103 equations, 20 figures, 2 tables, 2 algorithms)

This paper contains 29 sections, 7 theorems, 103 equations, 20 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Related work
  4. Sliced Wasserstein (SW) distance
  5. Hilbert curve projection (HCP) distance
  6. Binary space partitioning (BSP) distance
  7. Proposed method
  8. Recursive rank matching distance
  9. Statistical convergence of the anchored empirical RRM
  10. The last-mile phenomenon and SRRM
  11. Empirical RRM and the last-mile phenomenon
  12. SRRM algorithms and complexity
  13. Simulations
  14. Empirical validation of the last-mile plateau mechanism
  15. Experiment 1
  16. ...and 14 more sections

Key Result

Lemma 3.3

Let $T_\mu:[0,1]\to\mathbb{R}^d$ in Definition lem:conditional-holder1. Then $T_\mu$ is Borel measurable and satisfies $(T_\mu)_\# \mathrm{Unif}[0,1] = \mu.$

Figures (20)

  • Figure 1.1: The last-mile phenomenon of recursive partitioning methods.
  • Figure 2.1: (a) Samples of the source and target distributions. (b) Illustrations of how various distance metrics change as $\alpha$ increases. (c) Comparisons of different distance metrics in terms of runtime.
  • Figure 2.2: Hilbert space-filling curve Visualization.
  • Figure 2.3: Samples of the distributions and illustrations of how various distance metrics change.
  • Figure 2.4: Samples of the last-mile problem.
  • ...and 15 more figures

Theorems & Definitions (19)

  • Remark 3.1
  • Definition 3.2: Mass-median axis-recursive RRM tree curve and induced transport map
  • Lemma 3.3: Pushforward property of the induced map
  • Definition 3.4
  • Theorem 3.5
  • Lemma 3.7: Global Hölder control
  • Theorem 3.8: Consistency and rate for the anchored empirical RRM
  • Corollary 3.9: Two-sample stability of $\mathrm{RRM}$
  • Theorem 3.10: Finite-depth consistency of the empirical mass-median tree
  • Definition 4.1: Premature splitting under depth-$H$ address restriction
  • ...and 9 more