Fast distance computation of multivariate distributions via nonparanormal transport

TL;DR

This paper introduces the Nonparanormal Transport (NPT) metric, a fast and scalable method for computing distances between multivariate distributions, enabling efficient analysis in high-dimensional distributional data.

Contribution

The paper proposes the NPT metric, a closed-form, computationally efficient alternative to Wasserstein distance for multivariate distributions, suitable for large-scale data analysis.

Findings

NPT is at least 1000 times faster than existing Wasserstein variants.

NPT closely approximates Wasserstein distance in simulations.

Application to sleep study data demonstrates NPT's practical utility.

Abstract

With the increasing availability of data objects in the form of probability distributions, there is a growing need for statistical methods tailored to distributional data. Distance measures, especially the pairwise distance matrix between data objects, provide the foundation for a wide range of modern data analysis methods, such as clustering, multidimensional scaling, and distance-based regression, among others. The Wasserstein distance is commonly used with distributional data due to its compelling optimal transport property. However, while the Wasserstein distance can be efficiently computed for univariate distributions, its application to multivariate distributions is limited due to high computational costs. To address these scalability issues, we introduce the Nonparanormal Transport (NPT) metric, a closed-form distance based on the flexible nonparanormal distribution family for modeling skewed and non-Gaussian multivariate data. Simulation studies demonstrate that NPT maintains a high level of agreement with the Wasserstein distance, while being at least 1000 times faster than its efficient variants when computing a 100-distribution pairwise distance matrix in both 2 and 5 dimensions. We illustrate the utility of NPT through a multidimensional scaling analysis of bivariate oxygen desaturation distributions of 723 individuals with sleep apnea in the Sleep Heart Health Study.