Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives (2506.19025v1)

Abstract: In many applications of optimal transport (OT), the object of primary interest is the optimal transport map. This map rearranges mass from one probability distribution to another in the most efficient way possible by minimizing a specified cost. In this paper we review recent advances in estimating and developing limit theorems for the OT map, using samples from the underlying distributions. We also review parallel lines of work that establish similar results for special cases and variants of the basic OT setup. We conclude with a discussion of key directions for future research with the goal of providing practitioners with reliable inferential tools.

• The paper introduces novel estimators with explicit risk bounds and minimax rates for optimal transport map inference and uncertainty quantification.
• It leverages theoretical constructs like Brenier’s theorem and central limit theorems to rigorously connect estimation error with Wasserstein metrics.
• The study highlights practical applications in unpaired sample inference, spatial association, and high-dimensional quantile analysis while addressing computational-statistical tradeoffs.

Statistical Inference for Optimal Transport Maps: Advances and Perspectives

This paper provides a comprehensive review of recent developments in the statistical inference of optimal transport (OT) maps, with a focus on estimation, limit theorems, and practical inferential tools. The authors systematically address both the theoretical underpinnings and the practical implications of OT map estimation, highlighting connections to statistical applications, computational methods, and open problems in the field.

Overview and Motivation

Optimal transport maps, which rearrange mass between probability distributions in a cost-minimizing fashion, have become central objects in modern statistics, machine learning, and applied sciences. The review emphasizes that, in many applications, the OT map itself—not just the transport cost or coupling—is the primary target of inference. This is particularly relevant in settings such as:

• Unpaired sample inference (e.g., single-cell genomics), where the OT map models the evolution of distributions over time without paired observations.
• Measures of spatial association, where OT-based functionals (e.g., the Optimal Transport Colocalization curve) provide robust alternatives to classical statistics like Ripley’s K.
• Numerical discretization error quantification, where understanding the statistical properties of OT maps informs the accuracy of computational solvers.
• Multivariate quantiles and ranks, where OT maps generalize univariate quantile functions and enable new nonparametric inference procedures.

Theoretical Foundations

The paper reviews the Monge and Kantorovich formulations of OT, emphasizing the role of Brenier’s theorem, which guarantees the existence and uniqueness of OT maps as gradients of convex functions under suitable conditions. The duality structure of OT is leveraged to construct estimators and analyze their statistical properties.

Key theoretical contributions discussed include:

• Minimax estimation rates for OT maps under various regularity conditions.
• Stability bounds that relate the estimation error of OT maps to the Wasserstein distance between estimated and true distributions.
• Limit theorems (e.g., pointwise and process-level central limit theorems) for OT map estimators, particularly in smooth settings.

Practical Estimation Strategies

The review details several classes of estimators for OT maps:

• Empirical OT map via matching: Solving the discrete OT problem between empirical measures, with out-of-sample extension via nearest neighbors. This estimator is simple, tuning-free, and minimax optimal under strong convexity and smoothness, but cannot exploit higher smoothness.
• Plugin estimators: Estimating the source and target distributions (e.g., via kernel density estimation) and computing the OT map between these estimates. This approach achieves faster rates under smoothness assumptions but is computationally intensive in high dimensions.
• Dual estimators: Solving empirical versions of the semi-dual problem, often parameterized by neural networks, with theoretical guarantees under appropriate regularity.
• Entropic OT estimators: Leveraging the computational and statistical benefits of entropic regularization, which enables parametric rates and tractable inference at the cost of introducing bias and smoothing.

The paper provides explicit risk bounds and convergence rates for these estimators, highlighting the dependence on dimension, smoothness, and the structure of the distributions.

Special Cases and Variants

The review discusses several settings where statistical inference for OT maps is particularly tractable:

• One-dimensional OT: The OT map reduces to the quantile-quantile function, and plugin estimators achieve parametric rates with explicit Bahadur representations and bootstrap-based inference.
• Gaussian case: Closed-form expressions for the OT map allow for high-dimensional inference, with rates depending on the effective rank of covariance matrices.
• Semi-discrete and discrete OT: When one or both distributions are discrete, estimation rates become dimension-free, and central limit theorems are available for empirical OT maps and couplings.
• Entropic and divergence-regularized OT: Fixed regularization yields parametric rates and valid confidence bands, with recent work extending these results to Tsallis and other $f$-divergence regularizations, which can induce sparsity in the coupling.

A notable highlight is the Lower Complexity Adaptation (LCA) principle, which asserts that the minimax rate for OT estimation is governed by the lower intrinsic dimension of the two distributions, significantly mitigating the curse of dimensionality in certain regimes.

Limitations and Open Problems

The authors identify several limitations and open questions:

• Regularity assumptions: Most minimax and limit results require strong smoothness and convexity, which are not always satisfied in practice. The stability of OT maps under weaker conditions remains an active area of research.
• Computational-statistical tradeoffs: There is a need for joint analysis of computational complexity and statistical accuracy, especially for plug-in and dual estimators in high dimensions.
• Inference in low-regularity settings: Developing limit theorems and confidence sets for OT maps without strong regularity remains challenging.
• Tuning parameter selection: For entropic and divergence-regularized OT, principled methods for selecting regularization parameters are underdeveloped.
• Extension to alternative transport maps: The statistical theory for alternatives to Brenier maps (e.g., Knothe-Rosenblatt, diffusion models, rectified flow) is still nascent.

Implications and Future Directions

The reviewed advances have significant implications for both theory and practice:

• Statistical inference for OT maps is now feasible in a range of settings, enabling uncertainty quantification and hypothesis testing in applications from genomics to physics.
• Computational methods such as the Sinkhorn algorithm and neural network parameterizations are supported by emerging statistical guarantees.
• Dimension-adaptive rates and parametric inference in special cases expand the applicability of OT-based methods to high-dimensional and structured data.

Future work is expected to focus on relaxing regularity assumptions, developing computationally efficient and statistically optimal estimators, and extending the theory to broader classes of transport maps and cost functions. The interplay between statistical optimality, computational tractability, and practical relevance will continue to drive research in this area.