Geodesic and Adapted Exponential Smoothing: Wasserstein Smoothing for Multivariate Distributions and the Laws of Stochastic Processes

Abstract

Wasserstein exponential smoothing (WES) forecasts distributional time series on the line by lifting classical exponential smoothing to a geodesic in the 2-Wasserstein space. We recast its update as a θ-weighted Wasserstein barycenter and use this coordinate-free form to extend WES to multivariate distributions (closed form on Gaussians via Bures–Wasserstein), to laws of stochastic processes (via the adapted/bicausal distance), and to general divergences. A Hilbert meta-theorem shows the stationarity, mixing, and consistency guarantees transfer to any smoother linear in a Hilbert embedding, with a bounded-geometry refinement for curved targets. We then prove a central limit theorem for the smoothing parameter (asymptotic variance θ*(2−θ*)/d in the isotropic case), develop stable damped-trend and seasonal smoothers, identify WES with a steady-state Kalman filter and a Koopman resolvent, and extend the geometry to Finsler spaces—showing the quadratic cost is the unique flat point. Proofs and experiments are provided.

Keywords