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The fast rate of convergence of the smooth adapted Wasserstein distance

Martin Larsson, Jonghwa Park, Johannes Wiesel

TL;DR

This work tackles the slow convergence of empirical path-space distributions under the adapted Wasserstein distance by incorporating Gaussian smoothing, yielding a dimension-free fast rate. The authors show that for subgaussian $\mu$ and smoothing level $\sigma$, the smooth adapted Wasserstein distance satisfies $\mathbb{E}[\mathcal{A}\mathcal{W}^{(\sigma)}_p(\hat{\mu}_n, \mu)] \le C/\sqrt{n}$, by proving that smoothed measures have locally Lipschitz kernels and leveraging a dynamic programming principle together with empirical-process theory. Consequently, the ordinary Wasserstein distance inherits improved rates via $\mathcal{W}_p \le C\mathcal{A}\mathcal{W}_p$, under the same moment conditions, and the results extend to practical testing/statistical tasks such as SMPD-based martingale tests. The findings provide dimension-robust guarantees for time-dependent probability measures, with explicit dependence on $p$, $T$, $d$, $\sigma$, and exponential moment parameters. This advances the understanding of how smoothing interacts with adapted structures to overcome the COD in path-space estimation and related decision problems.

Abstract

Estimating a $d$-dimensional distribution $μ$ by the empirical measure $\hatμ_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hatμ_n, μ)]\lesssim n^{-1/d}$ for $d>2p$, where $\mathcal{W}_p$ denotes the $p$-Wasserstein metric. An effective tool to combat this curse of dimensionality is the smooth Wasserstein distance $\mathcal{W}^{(σ)}_p$, which measures the distance between two probability measures after having convolved them with isotropic Gaussian noise $\mathcal{N}(0,σ^2\text{I})$. In this paper we apply this smoothing technique to the adapted Wasserstein distance. We show that the smooth adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p^{(σ)}$ achieves the fast rate of convergence $\mathbb{E}[\mathcal{A}\mathcal{W}_p^{(σ)}(\hatμ_n, μ)]\lesssim n^{-1/2}$, if $μ$ is subgaussian. This result follows from the surprising fact, that any subgaussian measure $μ$ convolved with a Gaussian distribution has locally Lipschitz kernels.

The fast rate of convergence of the smooth adapted Wasserstein distance

TL;DR

This work tackles the slow convergence of empirical path-space distributions under the adapted Wasserstein distance by incorporating Gaussian smoothing, yielding a dimension-free fast rate. The authors show that for subgaussian and smoothing level , the smooth adapted Wasserstein distance satisfies , by proving that smoothed measures have locally Lipschitz kernels and leveraging a dynamic programming principle together with empirical-process theory. Consequently, the ordinary Wasserstein distance inherits improved rates via , under the same moment conditions, and the results extend to practical testing/statistical tasks such as SMPD-based martingale tests. The findings provide dimension-robust guarantees for time-dependent probability measures, with explicit dependence on , , , , and exponential moment parameters. This advances the understanding of how smoothing interacts with adapted structures to overcome the COD in path-space estimation and related decision problems.

Abstract

Estimating a -dimensional distribution by the empirical measure of its samples is an important task in probability theory, statistics and machine learning. It is well known that for , where denotes the -Wasserstein metric. An effective tool to combat this curse of dimensionality is the smooth Wasserstein distance , which measures the distance between two probability measures after having convolved them with isotropic Gaussian noise . In this paper we apply this smoothing technique to the adapted Wasserstein distance. We show that the smooth adapted Wasserstein distance achieves the fast rate of convergence , if is subgaussian. This result follows from the surprising fact, that any subgaussian measure convolved with a Gaussian distribution has locally Lipschitz kernels.

Paper Structure

This paper contains 12 sections, 15 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Adapted distances
  3. Approximations in adapted distances
  4. Prior results for the smooth adapted Wasserstein distance
  5. Main results
  6. Applications
  7. Notation and preparations
  8. Proof of Theorem \ref{['thm:fast_rate']}
  9. Kernels of smoothed measures
  10. Dynamic programming principle
  11. Empirical process theory
  12. The fast rate is sharp

Key Result

Theorem 4

Let $1<p<\infty$ and $\sigma>0$. Suppose that $\mu$ is a probability measure on $(\mathbb{R}^d)^T$, where $d\ge 1$ and $T\ge 2$, such that $\int e^{q\left\vert x\right\vert^2/(2\sigma^2)}\mu(dx)<\infty$ for $q>8p(2p-1)(T+9)$. Then there exists a constant $C>0$ that depends only on $p, q, d, T, \sigm

Figures (1)

  • Figure 1: $\mu=\frac{1}{2}\delta_{(0,1)}+\frac{1}{2}\delta_{(0,-1)}$ on the left and $\mu_{\varepsilon}=\frac{1}{2}\delta_{(\varepsilon, 1)}+\frac{1}{2}\delta_{(-\varepsilon, -1)}$ on the right.

Theorems & Definitions (32)

  • Definition 1: Smooth Wasserstein distance
  • Definition 2: Bicausal coupling
  • Definition 3: The adapted Wasserstein distance
  • Theorem 4: Fast rate
  • Proposition 5: Smoothed measures have Lipschitz kernels; exact statement in Proposition \ref{['prop:lipkernel']}
  • Proposition 6: Kernels of a smoothed measure
  • Proposition 7: Proposition $2.1$ in goldfeld2024limit
  • Remark 8
  • Lemma 9
  • proof
  • ...and 22 more