Entropic adapted Wasserstein distance on Gaussians

TL;DR

This paper extends the entropic adapted Wasserstein distance to multidimensional Gaussian processes, providing closed-form solutions for more efficient computation of distributional uncertainty in stochastic optimization.

Contribution

It generalizes previous work by deriving closed-form solutions for multidimensional Gaussian processes with entropic regularization, enhancing computational efficiency.

Findings

Closed-form solutions for multidimensional Gaussian processes.

Efficient computation of entropic adapted Wasserstein distance.

Extension of previous univariate results.

Abstract

The adapted Wasserstein distance is a metric for quantifying distributional uncertainty and assessing the sensitivity of stochastic optimization problems on time series data. A computationally efficient alternative to it, is provided by the entropically regularized adapted Wasserstein distance. Suffering from similar shortcomings as classical optimal transport, there are only few explicitly known solutions to those distances. Recently, Gunasingam--Wong provided a closed-form representation of the adapted Wasserstein distance between real-valued stochastic processes with Gaussian laws. In this paper, we extend their work in two directions, by considering multidimensional ($\mathbb{R}^d$-valued) stochastic processes with Gaussian laws and including the entropic regularization. In both settings, we provide closed-form solutions.