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

[Mat+26] introduced Wasserstein exponential smoothing (WES), a one-parameter recursiveforecaster for distributional time series on R, by lifting the classical exponential-smoothingrecursion to a geodesic interpolation in the 2-Wasserstein space and exploiting the isometrybetween W2(R) and the Hilbert space L2((0, 1)) of quantile functions. We develop threestructural generalizations and a unifying theory. First, we recast the WES update as a θ-weightedWasserstein barycenter, equivalently a point on a metric geodesic, and show that this variationalform (i) recovers WES verbatim in one dimension, (ii) is well posed on general nonpositivelycurved and barycentric metric spaces, and (iii) admits a closed form on multivariate Gaussiansvia the Bures–Wasserstein geometry, thereby resolving the multivariate extension flagged asopen in the original work. Second, we treat the case in which each observation is the law of astochastic process: we show that the naive Wasserstein geometry on path space is the wrongobject—value functionals from optimal stopping and hedging are discontinuous in it—and thatthe correct geometry is the adapted (bicausal) Wasserstein distance. Using the recent resultthat the adapted Wasserstein space is geodesic and supports barycenters, we define adaptedWES and give a backward-induction recursion for its update. Third, we place WES inside afamily of proximal/divergence smoothers μt = arg minρ(1 − θ)D(ρ∥μt−1) + θD(ρ∥νt), recoveringgeometric (log-linear) pooling, mixture smoothing, and Sinkhorn/sliced variants as specialcases, and clarifying the “mix versus morph” distinction. A Hilbert meta-theorem shows thatwhenever a smoother is linear in a Hilbert embedding, the stationarity, autocovariance-decay,and consistency results of WES transfer verbatim, and a bounded-geometry refinement carriesthem to the curved cases with explicitly degraded constants. Building on this theory we thenresolve the questions left open by the original work: we prove a central limit theorem for thesmoothing parameter, with√T(ˆθT − θ∗) → N(0, θ∗(2 − θ∗)/d) in the isotropic case; we give astable damped-trend and seasonal calculus; we identify WES with a steady-state Kalman filterand supply a convergent online estimator; and we cast it in Koopman operator form. Finally wemove the geometry beyond the Riemannian world: since the p-Wasserstein space is Finsler forp ̸= 2, a Banach/Lp meta-theorem shows the guarantees persist under two-uniform convexityand smoothness, degrading by constants that equal one precisely when p = 2—explaining theprivileged status of the quadratic cost—while a Berwald–Hadamard contraction handles thecurved Finsler case and identifies the abstract bounded-geometry modulus with Finsler anisotropy(the Cartan tensor). We give complete proofs and Bures–Wasserstein experiments confirmingboth consistency of the minimum-Wasserstein estimator in R2 and the predicted asymptoticvariance, and we close with merits and limitations.