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Entropic adapted Wasserstein distance on Gaussians

Beatrice Acciaio, Songyan Hou, Gudmund Pammer

TL;DR

The paper extends the closed-form solution for the entropic adapted Wasserstein distance to multidimensional Gaussian time-series, deriving an explicit formula for $\mathcal{AW}_{2,\lambda}^2$ between Gaussian laws and proving that the optimal coupling is Gaussian and unique when $\lambda>0$. It provides a complete characterization of Gaussian bi-causal couplings, shows when Monge maps exist, and uses a dynamic programming approach to establish the Gaussian optimizer property. The results illuminate differences between adapted and classical regularized transport, establish a tractable framework for high-dimensional time-series reliability assessment, and offer robust comparisons via entropic regularization, with applications in distributional uncertainty and stochastic optimization. Overall, the work delivers closed-form solutions, optimizer structure, and insightful comparisons that enhance computational efficiency and theoretical understanding of entropic adapted transport in Gaussian settings.

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.

Entropic adapted Wasserstein distance on Gaussians

TL;DR

The paper extends the closed-form solution for the entropic adapted Wasserstein distance to multidimensional Gaussian time-series, deriving an explicit formula for between Gaussian laws and proving that the optimal coupling is Gaussian and unique when . It provides a complete characterization of Gaussian bi-causal couplings, shows when Monge maps exist, and uses a dynamic programming approach to establish the Gaussian optimizer property. The results illuminate differences between adapted and classical regularized transport, establish a tractable framework for high-dimensional time-series reliability assessment, and offer robust comparisons via entropic regularization, with applications in distributional uncertainty and stochastic optimization. Overall, the work delivers closed-form solutions, optimizer structure, and insightful comparisons that enhance computational efficiency and theoretical understanding of entropic adapted transport in Gaussian settings.

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 (-valued) stochastic processes with Gaussian laws and including the entropic regularization. In both settings, we provide closed-form solutions.

Paper Structure

This paper contains 7 sections, 9 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations.
  3. Main results
  4. Adapted vs classical (regularized) Wasserstein distance
  5. On Monge optimizers of adapted Wasserstein distance
  6. Proof of \ref{['thm.opt_eaot_gau']}
  7. Examples

Key Result

Theorem 1.1

Let $\mu = \mathcal{N}(a,A)$ and $\nu = \mathcal{N}(b,B)$ be non-degenerate Gaussians on $\mathbb{R}^T$, whose covariance matrices have Cholesky decompositions $A = LL^\top$ and $B = MM^\top$. Then where $P_\lambda = f_\lambda(N)$For $f\colon \mathbb{R}\to \mathbb{R}$ and $M\in\mathbb{R}^{d \times d}$, we denote by $f(M)$ the element-wise function application of $f$ on $M$. and $N = \mathrm{diag}

Figures (2)

  • Figure 1: Different optimal couplings of $\mathcal{AW}_{2}$
  • Figure 2: Optimal couplings for different distances

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: Characterization of Gaussian bi-causal couplings
  • proof
  • Theorem 2.3: Existence of Gaussian bi-causal optimizers
  • Theorem 2.4: Closed-form representation
  • Theorem 2.5: Characterization of optimizers
  • proof : Proof of Theorems \ref{['thm.EAW_dT']}-\ref{['thm.EAW_dT_sol']}
  • Lemma 3.1
  • ...and 10 more