Extension of coupling via the Projection of Optimal Transport

TL;DR

This paper introduces a nonparametric method that combines limited coupled data with larger marginal datasets using optimal transport projection, improving statistical analysis with geometric and computational advantages.

Contribution

It extends coupling via optimal transport projection, providing a stable, efficient, and theoretically grounded approach to integrate coupled and marginal data.

Findings

Estimator has a natural geometric interpretation.

Sample complexity is derived using recent optimal transport theory.

Method can be approximated efficiently with entropic shadow.

Abstract

In many statistical settings, two types of data are available: coupled data, which preserve the joint structure among variables but are limited in size due to cost or privacy constraints, and marginal data, which are available at larger scales but lack joint structure. Since standard methods require coupled data, marginal information is often discarded. We propose a fully nonparametric procedure that integrates decoupled marginal data with a limited amount of coupled data to improve the downstream analysis. The approach can be understood as an extension of coupling via projection in optimal transport. Specifically, the estimator is a solution for the optimal transport projection over the space of probability measures, which genuinely provides a natural geometric interpretation. Not only is its stability established, but its sample complexity is also derived using recent advances in statistical optimal transport. In addition to this, we present its explicit formula based on ``shadow," a notion introduced by Eckstein and Nutz. Furthermore, the estimator can be approximated in almost linear time and in parallel by entropic shadow, which demonstrates the theoretical and practical strengths of our methods. Lastly, we present experiments with real and synthetic data to justify the performance of our method.