Wasserstein Distance Indices

• Wasserstein Distance Indices are quantitative metrics based on the optimal transport distance that compare probability measures by incorporating geometric and statistical properties.
• They include test statistics, proxies, and dependence coefficients, using methods like sliced and projected indices to offer computationally efficient approximations.
• These indices are applied in hypothesis testing, clustering, and model evaluation across imaging, Bayesian nonparametrics, and graph analysis, enhancing practical decision-making.

Wasserstein distance indices are quantitative summaries and derived metrics built on the Wasserstein transport distance, widely used to compare and analyze probability measures in mathematical statistics, optimal transport, computer science, and related fields. These indices include test statistics, proxies, dependence coefficients, and efficient computational surrogates, each exploiting the geometric, statistical, or operational structure of the Wasserstein metric. This article surveys core theoretical formulations, relaxations, surrogate indices, statistical properties, practical applications, and their implementation as developed in the literature.

1. Foundational Definitions and Theoretical Basis

The $p$-Wasserstein distance on a Polish metric space $(\mathcal X, d)$ between measures $\mu,\nu$ with finite $p$th moments is defined by

$$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$

where $\Gamma(\mu,\nu)$ denotes the set of couplings of $\mu$ and $\nu$. For $p=1$, the dual formulation is given by

$$W_1(\mu, \nu) = \sup_{f \in \operatorname{Lip}_1(\mathcal X)} \left\{\int f\, d\mu - \int f\, d\nu\right\},$$

with $(\mathcal X, d)$0 the class of 1-Lipschitz functions. Fundamental properties include nonnegativity, symmetry, the triangle inequality, and metrization of weak convergence plus moment convergence for the probability space $(\mathcal X, d)$1 (Panaretos et al., 2018).

Wasserstein distances seamlessly incorporate the geometry of the domain, which is central to their utility as indices in shape analysis, imaging, and probabilistic modeling.

2. Statistical Wasserstein Indices: Estimation and Testing

Wasserstein distances play a pivotal role as statistical indices for hypothesis testing, clustering, and model evaluation:

2.1. Empirical Estimation

The plug-in estimator $(\mathcal X, d)$2, where $(\mathcal X, d)$3 are empirical measures from i.i.d. samples, satisfies $(\mathcal X, d)$4 almost surely (rates are dimension-dependent; $(\mathcal X, d)$5 for $(\mathcal X, d)$6 and $(\mathcal X, d)$7 for discrete/finitely supported laws) (Panaretos et al., 2018).

2.2. Goodness-of-fit and Clustering

• Goodness-of-fit: $(\mathcal X, d)$8 is used for one-sample tests.
• Two-sample testing: $(\mathcal X, d)$9 serves as a nonparametric test statistic.
• Clustering: Minimizing within-cluster Wasserstein dispersion yields robust clustering procedures (Panaretos et al., 2018).

2.3. Disparity and Inequality

The Wasserstein–Gini index is defined via the $\mu,\nu$0 distance between quantile functions: $\mu,\nu$1 (Panaretos et al., 2018).

2.4. Explicit Null-Model Indices

On the simplex $\mu,\nu$2, with $\mu,\nu$3 computed via the cumulative coordinate trick, exact closed-form moments are available: $\mu,\nu$4 characterizing the typical scale and variability when comparing random discrete distributions (Frohmader et al., 2019).

3. Relaxation, Surrogate, and Proxy Indices

To address computational or statistical challenges, several effective indices serve as proxies for exact Wasserstein distances.

3.1. Sliced and Projected Indices

• Sliced Wasserstein ($\mu,\nu$5): Defined as $\mu,\nu$6, enjoying dimension-free sample complexity and sub-Gaussian concentration (Xu et al., 2022).
• Maximum Sliced Wasserstein ($\mu,\nu$7): $\mu,\nu$8, with uniform tail bounds and nonparametric Donsker-theorem-based CLTs (Xu et al., 2022).
• Observable Wasserstein Distance: $\mu,\nu$9, with computationally tractable lower-bounding pseudo-metrics $p$0 given by maximizing over a finite anchor-set hierarchy. These proxies form a tunable hierarchy trading sharpness for efficiency (Santos et al., 11 May 2026).
• Distance-Matrix Wasserstein ($p$1): For metric-measure spaces, compares the distribution of finite random distance matrices; $p$2 (GW = Gromov–Wasserstein), converging to GW as $p$3, with tight finite-sample and dimension-adaptive bounds (Xu et al., 14 May 2026).

3.2. Proxies via Discrepancies and KS Distance

On $p$4, $p$5 can be sharply upper bounded by powers of box discrepancies: $p$6 where $p$7 is the uniform box discrepancy; or, in $p$8, by KS distance-weighted moments: $p$9 offering practical computable indices for model assessment and sampling (Pagès et al., 5 May 2026).

4. Indices in Complex Structures: Dependence and Graphs

4.1. Dependence Indices

• Wasserstein Correlation Coefficient: For a coupling $$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$0 between $$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$1,

$$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$2

which is $$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$3 at independence ($$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$4), $$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$5 under functional dependence, convex in $$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$6, and admits nonparametric plug-in estimation and independence testing procedures (Wiesel, 2021).

• Wasserstein Index of Dependence for random measures: $$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$7, quantifies the dependence structure within vectors of completely random measures, normalizing the Wasserstein distance between full dependence and independence. This index is intrinsically multivariate, nonpairwise, numerically tractable, and directly enables prior specification and model selection in Bayesian nonparametrics (Catalano et al., 2021).

4.2. Graph, Matrix, and Kernel-Induced Indices

• Graph Distance via GMMs: Nodes of graphs are mapped to (probabilistic) embeddings, fitted as Gaussian mixtures; the resulting Wasserstein distance between mixtures quantifies graph dissimilarity and admits fast closed-form or OT-based computation under tied/diagonal covariance structure, scaling to large graphs (Scholkemper et al., 2024).
• Tree-Wasserstein Distance (Supervised): For document or feature-collection comparison, fast computation uses parent–child summation formulas on trees, with end-to-end differentiable relaxations for metric learning, outperforming exact OT on large corpora (Takezawa et al., 2021).
• Quasi-Manhattan Wasserstein Distance: Combines three 1D Wasserstein computations on linearized, rotated, and transposed matrix representations, providing a linear-time, sub-5% error proxy for high-dimensional matrix data (Lim, 2023).
• Kernel Wasserstein: Embeds measures into a reproducing kernel Hilbert space; $$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$8 serves as a fast, closed-form Wasserstein-type index, enabling robust distributional comparisons in applications like imaging and anomaly detection (Oh et al., 2019).

5. Generalizations, Unbalanced Cases, and Dualities

Classical Wasserstein distances are only defined for equal total mass. The generalized Wasserstein distance

$$W_p(\mu,\nu) = \left(\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathcal X\times \mathcal X} d(x, y)^p\, d\gamma(x, y)\right)^{1/p},$$9

admits source terms (mass creation/destruction) at cost $\Gamma(\mu,\nu)$0, and transport at cost $\Gamma(\mu,\nu)$1, remaining a metric for all nonnegative measures. In special cases, e.g. $\Gamma(\mu,\nu)$2, it coincides with the flat (bounded–Lipschitz) distance, providing a dual characterization built on test functions bounded both in norm and Lipschitz seminorm (Piccoli et al., 2013).

The corresponding dynamic (Benamou–Brenier) formulations include penalties for both kinetic energy and total source mass, extending applicability to unbalanced data and source-driven PDEs.

6. Numerical Methods, Concentration, and Algorithmic Aspects

Efficient computation of Wasserstein indices leverages diverse algorithmic approaches:

• Exact OT: Solved via linear programming or the Sinkhorn (entropic regularization) algorithm, with respective $\Gamma(\mu,\nu)$3 and $\Gamma(\mu,\nu)$4 complexity for $\Gamma(\mu,\nu)$5 support points.
• Sliced, Observable, and DMW: Lower computational cost via random projections or anchor-based subspaces; error decays as $\Gamma(\mu,\nu)$6 for $\Gamma(\mu,\nu)$7 projections.
• Tree/Supervised/KD-tree/Quasi-linear proxies: For document, image, or graph comparisons, sacrifices exactness for scalability with proven $\Gamma(\mu,\nu)$8 or empirical error guarantees (Takezawa et al., 2021, Lim, 2023, Xu et al., 14 May 2026).
• Statistical consistency: Many indices admit nonparametric CLTs, sub-Gaussian tail inequalities, and explicit rate guarantees; rates can be optimal or dimension-dependent, highlighting trade-offs for high-dimensional data (Xu et al., 2022, Santos et al., 11 May 2026).

7. Applications and Impact in Practice

Wasserstein distance indices are pervasive across theoretical and applied contexts:

• Hypothesis testing: Two-sample and independence tests in high-dimensional and complex domains (Panaretos et al., 2018, Wiesel, 2021).
• Fair benchmarking: Using scaled or normalized Wasserstein indices as null distributions or performance thresholds for random or expected scenarios (Frohmader et al., 2019).
• Model selection: Informativity and dependence control in structured Bayesian models via dependence indices (Catalano et al., 2021).
• Computational geometry and imaging: Proxies (sliced, observable, kernel) drive applications in anomaly detection, image retrieval, generative modeling, and shape analysis (Santos et al., 11 May 2026, Oh et al., 2019).
• Numerical methods and QMC: Proxy indices facilitate efficient error control in sampling, approximation, and integration, e.g., via Proinov’s theorem or box-discrepancy bounds (Pagès et al., 5 May 2026).

In sum, Wasserstein distance indices, in their many forms, provide a fundamental and flexible toolbox for quantifying, comparing, and understanding probability laws, structures, and models in modern applied mathematics and statistics. Their evolving proxies, computational relaxations, and derivative indices address the inherent complexity of the metric, enabling robust analysis, inference, and decision-making across diverse domains.