Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications

TL;DR

This paper introduces the linear barycentric coding model (LBCM) using the LOT metric for analyzing and synthesizing probability measures, providing theoretical insights, computational methods, and applications in covariance estimation and data imputation.

Contribution

The paper presents a closed-form solution for LBCM, establishes its equivalence to 2-Wasserstein barycenters for compatible measures, and demonstrates its capacity to represent all measures on the unit interval.

Findings

LBCM provides a closed-form solution for measure analysis and synthesis.

LBCM is equivalent to 2-Wasserstein barycenters for compatible measures.

LBCM effectively aids in covariance estimation and data imputation.

Abstract

We propose the linear barycentric coding model (LBCM) which utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the closed unit interval. We show that a natural analogous construction of an LBCM in 2 dimensions fails, and we leave it as an open problem to identify the proper extension in more than 1 dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.