---
title: Empirical Wasserstein-1 Distance Overview
url: https://www.emergentmind.com/topics/empirical-wasserstein-1-distance
type: topic
---

# Empirical Wasserstein-1 Distance Overview

The empirical Wasserstein-1 distance, also known as the empirical earth mover’s distance, quantifies the discrepancy between empirical and population distributions or between two empirical distributions via optimal transport with linear cost. This metric plays a central role in probability, statistical inference, machine learning, and high-dimensional data analysis due to its mathematical tractability and direct connection to geometry and coupling. Both the theory and practice of empirical Wasserstein-1 distance are well-developed, with precise asymptotics, non-asymptotic deviation bounds, and efficient computation/accessibility in various settings.

## 1. Formal Definition and One-Dimensional Characterization

For probability measures $\mu, \nu \in \mathcal{P}_1(E)$ on a Polish metric space $(E,d)$, the 1-Wasserstein distance is defined as:
\[
W_{1}(\mu,\nu) = \inf_{\gamma \in \Gamma(\mu,\nu)} \int_{E \times E} d(x,y) \,\gamma(dx,dy)
\]
where $\Gamma(\mu, \nu)$ is the set of all couplings of $\mu$ and $\nu$. The Kantorovich-Rubinstein duality gives:
\[
W_1(\mu, \nu) = \sup_{\|f\|_{\text{Lip}} \leq 1} \left| \int f\, d\mu - \int f\, d\nu \right|
\]
For empirical measures $\mu_n = \frac{1}{n} \sum_{i=1}^n \delta_{X_i}$, $X_i \sim \mu$, $W_1(\mu_n, \mu)$ captures the optimal cost of transporting the empirical distribution to the true law.

On $\mathbb{R}$, a fundamental property is:
\[
W_1(F, G) = \int_{-\infty}^\infty |F(x) - G(x)|dx = \int_0^1 |F^{-1}(u) - G^{-1}(u)| du
\]
for cumulative distribution functions $F, G$, and quantile functions $F^{-1}, G^{-1}$ [2111.03570]. This quantile formula underpins both practical computation (requiring only sorting) and theoretical analysis.

## 2. Asymptotic and Finite-Sample Rates of Convergence

The rate at which $W_1(\mu_n, \mu) \rightarrow 0$ as $n \to \infty$ depends on the geometry of the support and moment conditions.

- **General compact metric spaces**: If the support has dimension $d$ (in a suitable sense), then [1707.00087, 2508.02615, 1312.2128]:
  \[
  \mathbb{E}[W_1(\mu_n, \mu)] \asymp
  \begin{cases}
  n^{-1/d}, & d > 2 \\
  n^{-1/2} \log n, & d=2 \\
  n^{-1/2}, & d < 2
  \end{cases}
  \]
  The optimality of these rates is established via dyadic partition coupling and metric entropy arguments.
- **Moment conditions**: With finite moment of order $q > 1$, these rates hold, with an additional $O(n^{-(q-1)/q})$ term when $q$ is not large enough for the dimension [1312.2128, 1807.09031].
- **Sub-Gaussian and exponential tails**: Yield exponential deviations and matching (up to constants) moment bounds [1902.10709, 1312.2128].
- **Singular measures or low-dimensional support**: Intrinsic dimension determines the empirical convergence rate. The theory covers pre-asymptotic regimes where effective dimensionality is lower than the ambient [1707.00087, 2508.02615].

## 3. Limit Distributions and Weak Convergence

In dimension one with smooth density and regular tails, the plug-in statistic $W_1(F_n, G_n)$ satisfies functional central limit theorems:
- **Two-sample case** $(F \neq G)$: Under regularity, [1911.02389] proves
  \[
  \sqrt{n}\left( W_1(F_n, G_n) - W_1(F, G) \right) \Rightarrow \mathcal{N}(0, \sigma^2),
  \]
  where $\sigma^2$ is an explicit quadratic form involving Brownian bridges and the quantile process.

- **Goodness-of-fit case** $(F = G)$: The standard $\sqrt{n}$-CLT fails. The limiting distribution is non-Gaussian and the scaling rate is slower (controlled by the regular variation at zero of the cost function $\rho(x) = |x|$), specifically
  \[
  n^{1/2}/\log n \cdot W_1(F_n, F) \Rightarrow \int_0^1 | \mathcal{B}(u) | du,
  \]
  where $\mathcal{B}$ is a standard Brownian bridge [1911.02389].

- **Finite metric spaces**: For discrete support, $W_1$ is the value of a random linear program. The asymptotic distribution is the maximum of linear forms in a Gaussian random vector over the dual constraint set, yielding a non-classical limit; naive bootstrap fails, and valid alternatives are derived [1610.03287].

- **General dimensions**: Functional delta method and empirical process theory remain technically challenging; for $d \geq 2$, strong regularity is often required for CLTs.

## 4. Non-Asymptotic Deviation, Concentration, and Sample Complexity

Sharp deviation inequalities for $W_1(\mu_n, \mu)$ are available under moment or transport-entropy assumptions:
- **Deviation bounds**: If $\mu$ is sub-Gaussian, for any $\epsilon > 0$,
  \[
  \Pr( W_1(\mu_n, \mu) \geq \epsilon ) \leq C \exp( -c n \epsilon^2 )
  \]
  with explicit constants. For sub-exponential and heavy-tailed distributions, similar, though possibly polynomial, concentration rates hold [1902.10709, 1103.3188, 1312.2128].

- **Transport-entropy inequalities**: If $\mu$ satisfies a $T_1(C)$-inequality (Gaussian-type concentration in $W_1$), McDiarmid's bounded differences yield for all $t > 0$,
  \[
  \mathbb{P}\bigl( W_1(\mu_n, \mu) \geq \mathbb{E}[W_1(\mu_n, \mu)] + t \bigr) \leq \exp\left( - n t^2 / (8C) \right)
  \]
  [1103.3188].

- **Sample complexity**: Achieving $W_1(\mu_n, \mu) \leq \epsilon$ with probability $1 - \delta$ requires $n \gtrsim \epsilon^{-2} \log(1/\delta)$ in sub-Gaussian settings [1902.10709].

## 5. Computational Methods and Approximations

Efficient practical and approximate computation of empirical $W_1$ is critical in large-scale applications:
- **1D exact computation**: $O(n)$ time via sorting and using either the empirical CDF or quantile formula [2111.03570].
- **Multivariate exact computation**: $W_1$ reduces to a linear program of size $n \times n$; complexity is typically $O(n^3)$.
- **Tree-based Approximation (TWD)**: Embeds the data in a tree metric and solves a convex Lasso problem (nonnegative $\ell_1$-regularized regression) for optimal edge weights, yielding linear-time approximate computation with quantifiable accuracy. Variance is reduced via tree-slicing (averaging over trees) [2206.12116].
- **Deep Network Approximation**: In high-dimensional settings, the Lipschitz function class is approximated by 1-Lipschitz neural networks; the supremum in the dual representation is optimized over networks, enabling scalable hypothesis tests and confidence intervals via Gaussian multiplier bootstrap [1910.07773].

| Method            | Dimension | Computational Cost                |
|-------------------|-----------|-----------------------------------|
| 1D Sort+Pairing   | 1         | $O(n\log n)$                      |
| LP Solver         | $d$       | $O(n^3)$ (network flow)           |
| Tree-Wasserstein  | $d$       | $O(N)$ (tree nodes)               |
| ReLU Network Dual | $d$       | $O(nT)$ (SGD/ADAM; $T$ bootstraps)|

## 6. Statistical Inference, Hypothesis Testing, and Confidence Bands

Empirical Wasserstein-1 distance underpins a variety of inference schemes:
- **Hypothesis testing**: Empirical $W_1$-based one- and two-sample tests with Gaussian process bootstrap calibration have correct Type I error and comparable or superior performance to alternatives, even on singular supports [1910.07773].
- **Confidence intervals**: Bootstrap quantiles of the supremum of Gaussian processes (approximating the empirical process indexed by 1-Lipschitz functions) yield valid CIs for $W_1$ and for functionals that are $W_1$-Lipschitz [1910.07773, 1610.03287, 1902.10709].
- **Applications**: Used to rigorously quantify inter/intra-group distances in metagenomics and other high-dimensional histogram data, providing interpretable intervals and robust significance estimates even in challenging regimes (e.g., partially overlapping supports) [1610.03287].

## 7. Optimality, Quantization, and Theoretical Extensions

Empirical measures are, up to polylogarithmic factors, as effective as optimal uniform quantizers for 1-Wasserstein approximation:
- **Quantization error**: The expected empirical Wasserstein-1 distance nearly matches the minimal error over all $n$-point uniform quantizers up to a factor $O(\log n)$, the gap being sharply characterized with multiscale decomposition, chaining arguments, and metric entropy bounds [2508.02615].
- **Non-uniform quantizers**: In many settings (e.g., absolutely continuous measures on $\mathbb{R}^{d}$), the empirical quantization rate matches optimal non-uniform quantizers. However, for measures with small-mass fine-structure, polynomial factors may appear [2508.02615].
- **Transport-entropy connections**: $W_1$-concentration unifies analysis of risk measures, generalizing CVaR and other quantile-related risk bounds to arbitrary $L$-Lipschitz functionals [1902.10709].
- **Open questions**: The necessity of polylogarithmic rate gaps in general spaces, precise characterization for $p$-Wasserstein, and performance for strongly singular or heavy-tailed distributions remain subjects of active research [2508.02615].

## References
- [2111.03570] Why the 1-Wasserstein distance is the area between the two marginal CDFs.
- [1911.02389] Weak convergence of empirical Wasserstein type distances.
- [1707.00087] Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance.
- [1312.2128] On the rate of convergence in Wasserstein distance of the empirical measure.
- [2508.02615] Optimality of empirical measures as quantizers.
- [1902.10709] A Wasserstein distance approach for concentration of empirical risk estimates.
- [1103.3188] Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance.
- [2206.12116] Approximating 1-Wasserstein Distance with Trees.
- [1910.07773] Hypothesis Test and Confidence Analysis with Wasserstein Distance on General Dimension.
- [1610.03287] Inference for Empirical Wasserstein Distances on Finite Spaces.
- [1807.09031] Behavior of the empirical Wasserstein distance in $\mathbb{R}^d$ under moment conditions.
- [2101.08126] A short proof on the rate of convergence of the empirical measure for the Wasserstein distance.

Source: https://www.emergentmind.com/topics/empirical-wasserstein-1-distance