Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications
This work addresses the challenge of efficiently representing and manipulating probability distributions for researchers in machine learning and statistics, though it is incremental as it builds on existing optimal transport frameworks.
The authors tackled the problem of analyzing and synthesizing probability measures by proposing the linear barycentric coding model (LBCM) using the linear optimal transport metric, providing a closed-form solution and establishing equivalence to 2-Wasserstein barycenters for compatible measures, with applications in covariance estimation and data imputation.
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.