Wasserstein Distance Approximations

• Wasserstein distance approximations are methods for estimating optimal transport metrics, balancing computational efficiency with theoretical error bounds.
• They leverage techniques like tree-based partitioning, entropic regularization, slicing projections, and neural network surrogates to reduce computation time.
• These methods enable scalable and robust estimations in high dimensions, with practical applications in statistical inference and machine learning.

The Wasserstein distance, or optimal transport metric, is a central tool to quantify discrepancies between probability distributions across a broad range of modern probabilistic and statistical domains. However, its computational bottleneck and the challenge of obtaining theoretically justified approximation and error bounds have given rise to a rich landscape of Wasserstein distance approximations. These approximations balance tractability, statistical guarantees, and scalability, relying on advanced concepts from optimization, measure concentration, kernel methods, robust statistics, neural architectures, and random geometric structures.

1. Fundamental Principles and Exact Formulations

The $p$-Wasserstein distance between probability measures $\mu,\nu$ on a metric space $(\mathcal{X},d)$ is defined as: $$W_p(\mu, \nu) = \left( \inf_{\pi \in \Pi(\mu, \nu)} \int_{\mathcal{X}\times\mathcal{X}} d(x, y)^{p} \, d\pi(x, y) \right)^{1/p}$$ where $\Pi(\mu, \nu)$ is the set of couplings with prescribed marginals. For empirical measures supported on $n$ samples each, computation reduces for $p=1$ to an assignment (matching) problem, but for $d>1$ this remains computationally intensive ($O(n^3)$ time for general cases). In one dimension, sorting yields an optimal coupling.

The design of Wasserstein approximations centers on either algorithmic acceleration (tree embeddings, entropic regularization, swap heuristics), statistical relaxation (sampling, quantization), problem-specific structure (Gaussian processes, covariance operators, hierarchical metrics), or function approximation (e.g., neural network surrogates).

2. Tree and Clustering-Based Discrete Approximations

Efficient distributional approximation with finite support and high-confidence error control is achieved by partitioning $\mathcal{X}$ and constructing a quantized empirical measure: $\mu,\nu$0 where $\mu,\nu$1 is a partition of $\mu,\nu$2, and $\mu,\nu$3 is the representative of each cell. The true distribution is unknown, but a high-confidence bound on $\mu,\nu$4 is provided by solving a tractable mixed-integer linear program (MILP) that maximizes the worst-case transport cost under empirical Clopper–Pearson-based confidence intervals for cell probabilities (Figueiredo et al., 8 Feb 2026).

Tree-based approaches use space-partitioning structures (quadtree, kd-tree) to accelerate the transport computations:

• Flowtree and kd-Flowtree: The optimal transport plan is solved recursively according to the tree’s hierarchical structure, reducing computation to near-linear in support size, with provable approximation bounds dependent only on support size and intrinsic dimension (Teshigawara et al., 19 Jan 2026).
• Persistence diagram algorithms: Randomly shifted quadtrees or modified flowtrees enable scalable Wasserstein computation between persistence diagrams, with error controlled up to $\mu,\nu$5 and runtime $\mu,\nu$6 for $\mu,\nu$7 diagram points and spread $\mu,\nu$8 (Chen et al., 2021).

3. Sliced and Projected Wasserstein Approximations

Sliced-Wasserstein (SW) distances estimate $\mu,\nu$9 by projecting distributions onto random directions $(\mathcal{X},d)$0 and averaging 1D Wasserstein distances: $(\mathcal{X},d)$1

Monte Carlo estimation samples $(\mathcal{X},d)$2 random directions; however, under high-dimensional concentration of measure, the empirical distribution of 1D projections converges to a Gaussian, enabling deterministic closed-form approximations. Specifically, for $(\mathcal{X},d)$3: $(\mathcal{X},d)$4 under weak dependence, with the approximation error vanishing polynomially in $(\mathcal{X},d)$5 (Nadjahi et al., 2021). Deterministic SW-approximations run in $(\mathcal{X},d)$6 time and outperform even moderate-sample Monte Carlo estimates in high dimensions.

Generalized SW (GSW) distances extend slices to nonlinear projections, e.g., polynomial or neural functions, and similar high-dimensional Gaussian-equivalence phenomena enable closed-form, moment-based deterministic approximations for polynomial and neural projections (Le et al., 2022).

4. Neural, Universal, and Metric-Learning Approximations

Recent advances exploit universal neural architectures respecting permutation and group invariance, crucial for point cloud or document OT:

• SFGI networks: Any sufficiently smooth Wasserstein-type distance on weighted point sets can be closely approximated by a neural architecture comprising (i) symmetric, group-invariant embedding, (ii) set-level transformation, and (iii) symmetric combination (Chen et al., 2023). Universal approximation at scale-independent model complexity is established for diverse group actions, and empirical results confirm superior generalization for unseen set sizes.
• Supervised Tree-Wasserstein: Tree metrics augmented with differentiable and learnable “parenthood” allow supervised optimization of Wasserstein distances, with efficient GPU implementations suitable for batch processing and task-specific metric adaptation (Takezawa et al., 2021).

5. Kernel, Covariance Operator, and Process-Based Approximations

For Gaussian measures and processes, $(\mathcal{X},d)$7 is computable via covariance operators: $(\mathcal{X},d)$8 Finite-sample estimators for the exact and Sinkhorn-regularized Wasserstein distances between Gaussian processes can be constructed via empirical Gram matrices or sample covariance operators, achieving $(\mathcal{X},d)$9 consistency for Sinkhorn regardless of the underlying dimension (Quang, 2021).

6. Robust, Federated, and High-Dimensional Wasserstein Approximations

• Robust MoM-based estimation: Median-of-means (MoM) procedures, extended to dual Kantorovich forms, yield estimators of $$W_p(\mu, \nu) = \left( \inf_{\pi \in \Pi(\mu, \nu)} \int_{\mathcal{X}\times\mathcal{X}} d(x, y)^{p} \, d\pi(x, y) \right)^{1/p}$$0 resilient to substantial adversarial contamination. Consistency and finite-sample error rates are established under sublinear outlier growth (Staerman et al., 2020).
• Federated computation: Iterative approaches leverage the triangle inequality and geodesic convexity to decompose Wasserstein computation across clients, with error decaying geometrically in iteration number, supporting privacy-preserving distributed estimation (Rakotomamonjy et al., 2023).
• Deterministic quantization: Optimal