Distributional Convergence of Empirical Entropic Optimal Transport and Statistical Applications

TL;DR

This paper establishes the asymptotic distribution of empirical Entropic Optimal Transport functionals, enabling statistical inference and confidence band construction, with applications in biological data analysis.

Contribution

It provides the first weak convergence results for a broad class of EOT functionals, including colocalization measures, using Hadamard differentiability and the delta method.

Findings

Derived asymptotic weak convergence for EOT functionals.

Constructed uniform confidence bands for colocalization curves.

Validated theory with simulations and real mitochondrial data.

Abstract

Recently, the statistical properties of empirical Entropic Optimal Transport (EOT) have attracted great interest, as this quantity has been shown to be useful for complex data analysis, among other reasons due to its computational efficiency. In several applications, it has been observed that the EOT plan provides valuable information beyond just the optimal value. For example, in cell biology, colocalization analysis based on the EOT plan has been introduced as a measure for quantification of spatial proximity of different protein assemblies. Despite recent progress in the analysis of its risk properties, a precise understanding of its statistical fluctuations to make it accessible for inference remains elusive to a large extent. In this paper, we derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method. As an application, we obtain uniform confidence bands for colocalization curves and bootstrap consistency. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.