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arXiv · 2610.38519

High-dimensional two-sample testing: From optimal transport to optimal derangements

Abhradipta Ghosh·Vaibhav Sherkar·Bilol Banerjee·Anil K. Ghosh
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Abstract

In recent years, optimal transport has emerged as a useful framework for constructing distribution-free multivariate two-sample tests by transporting observations to a reference distribution under a suitable transportation cost. However, existing optimal transport-based tests can perform poorly in high dimension, low sample size (HDLSS) settings, and their performance depends critically on the choice of the reference distribution and transportation cost. We investigate this phenomenon and show that suitable choices can substantially improve their performance. Motivated by this analysis, we introduce a class of two-sample tests based on optimal derangements that do not require an external reference distribution. The minimum distance pairing test of Rosenbaum arises as a special case when the optimization is restricted to a subclass of derangements, thereby revealing a connection between this method and optimal transport. We propose the optimal derangement (ODER) test and its distribution-free variant, the ODER-SHC test, obtained by restricting the minimization to derangements consisting of a single Hamiltonian cycle. We establish their consistency under both HDLSS and classical large-sample asymptotic regimes. We also develop corresponding procedures based on the $L1$ distance and establish their high-dimensional consistency under conditions involving coordinate-wise energy distances. These $L1$ distance based tests can also discriminate between two high-dimensional distributions with identical first two moments but different marginal distributions. Analyses of benchmark datasets and extensive simulation studies demonstrate the performance of the proposed methods across a variety of alternatives and data settings.

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Abhradipta Ghosh, Vaibhav Sherkar, Bilol Banerjee, Anil K. Ghosh. 2026-10-28. High-dimensional two-sample testing: From optimal transport to optimal derangements. https://arxiv.org/abs/2610.38519

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