Electronic Journal of Statistics

Central limit theorems for entropy-regularized optimal transport on finite spaces and statistical applications

Jérémie Bigot, Elsa Cazelles, and Nicolas Papadakis

Full-text: Open access

Abstract

The notion of entropy-regularized optimal transport, also known as Sinkhorn divergence, has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data analysis. The Sinkhorn divergence allows the fast computation of an entropically regularized Wasserstein distance between two probability distributions supported on a finite metric space of (possibly) high-dimension. For data sampled from one or two unknown probability distributions, we derive the distributional limits of the empirical Sinkhorn divergence and its centered version (Sinkhorn loss). We also propose a bootstrap procedure which allows to obtain new test statistics for measuring the discrepancies between multivariate probability distributions. Our work is inspired by the results of Sommerfeld and Munk in [33] on the asymptotic distribution of empirical Wasserstein distance on finite space using unregularized transportation costs. Incidentally we also analyze the asymptotic distribution of entropy-regularized Wasserstein distances when the regularization parameter tends to zero. Simulated and real datasets are used to illustrate our approach.

Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 5120-5150.

Dates
Received: February 2019
First available in Project Euclid: 12 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1576119711

Digital Object Identifier
doi:10.1214/19-EJS1637

Mathematical Reviews number (MathSciNet)
MR4041704

Zentralblatt MATH identifier
07147373

Subjects
Primary: 62G20: Asymptotic properties 62G10: Hypothesis testing 65C60: Computational problems in statistics

Keywords
Optimal transport Sinkhorn divergence central limit theorem bootstrap hypothesis testing

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bigot, Jérémie; Cazelles, Elsa; Papadakis, Nicolas. Central limit theorems for entropy-regularized optimal transport on finite spaces and statistical applications. Electron. J. Statist. 13 (2019), no. 2, 5120--5150. doi:10.1214/19-EJS1637. https://projecteuclid.org/euclid.ejs/1576119711


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