Conditional Wasserstein Barycenters and Interpolation/Extrapolation of Distributions

Abstract

Increasingly complex data analysis tasks motivate the study of the dependency of distributions of multivariate continuous random variables on scalar or vector predictors. Statistical regression models for distributional responses so far have primarily been investigated for the case of one-dimensional response distributions. We investigate here the case of multivariate response distributions while adopting the 2-Wasserstein metric in the distribution space. The challenge is that unlike the situation in the univariate case, the optimal transports that correspond to geodesics in the space of distributions with the 2-Wasserstein metric do not have an explicit representation for multivariate distributions. We show that under some regularity assumptions the conditional Wasserstein barycenters constructed for a geodesic in the Euclidean predictor space form a corresponding geodesic in the Wasserstein distribution space and demonstrate how the notion of conditional barycenters can be harnessed to interpolate as well as extrapolate multivariate distributions.The utility of distributional inter-and extrapolation is explored in simulations and examples.We study both global parametric-like and local smoothing-like models to implement conditional Wasserstein barycenters and establish asymptotic convergence properties for the corresponding estimates. For algorithmic implementation we make use of a Sinkhorn entropy-penalized algorithm. Conditional Wasserstein barycenters and distribution extrapolation are illustrated with applications in climate science and studies of aging.

Inference for Projection-Based Wasserstein Distances on Finite Spaces

“…Consequently, we simplify the regression of Gaussian distribution-on-Gaussian distribution to matrix-on-matrix linear regression. Given the explicit expression of the optimal transport map between Gaussian distributions as (7), our transformation avoids computational difficulties. Although our transformation is not isometric in general, it has certain isometric properties as shown in Proposition 1.…”

“…for some positive definite matrix A ∈ R d×d and vector v ∈ R d . According to Theorem 2.4 of [8], the closed-forms of the Wasserstein distance (6) and optimal transport map (7) are valid for any two measures µ 1 , µ 2 in the same class of elliptically symmetric distributions P f (R d ). Since our models rely only the forms ( 6), (7), our result can be extended to the case in which (ν 1 , ν 2 ) are…”

“…There are several advantages to using the Wasserstein metric: it gives more intuitive interpretations of mean and geodesics compared to other metrics, and it reduces errors by rigorously treating constraints as distribution functions. Based on this approach, numerous methods have been proposed for the anlaysis of distribution data ( [2,18,17,7,4,9,26]).…”

“…[17] developed regression models for coupled vector predictors and univariate random distributions as responses. [7] developed regression models for multivariate response distributions. [4] and [9] proposed regression models for scenarios where both regressors and responses are random distributions, and [10] studies its extension to the multivariate case.…”