Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimation

2026-02-04

AI Summary

This work addresses the challenge that discretization sampling error in large-scale optimal transport problems is fundamentally constrained by the intrinsic dimensionality of the data, severely limiting the accuracy and convergence rate of Wasserstein distance estimation. To overcome this, the authors propose a semi-dual functional estimator that bypasses explicit optimal transport computation and simultaneously enables efficient estimation of both the sampling error and the intrinsic dimension. Building on a multi-scale analysis of error decay, the method adaptively calibrates the Sinkhorn regularization strength and incorporates Richardson extrapolation to produce a debiased estimate. The resulting approach is parameter-free, does not require an OT solver, and achieves significantly improved estimation accuracy and convergence rates while maintaining computational efficiency.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Solving large scale Optimal Transport (OT) in machine learning typically relies on sampling measures to obtain a tractable discrete problem. While the discrete solver's accuracy is controllable, the rate of convergence of the discretization error is governed by the intrinsic dimension of our data. Therefore, the true bottleneck is the knowledge and control of the sampling error. In this work, we tackle this issue by introducing novel estimators for both sampling error and intrinsic dimension. The key finding is a simple, tuning-free estimator of $\text{OT}_c(\rho, \hat\rho)$ that utilizes the semi-dual OT functional and, remarkably, requires no OT solver. Furthermore, we derive a fast intrinsic dimension estimator from the multi-scale decay of our sampling error estimator. This framework unlocks significant computational and statistical advantages in practice, enabling us to (i) quantify the convergence rate of the discretization error, (ii) calibrate the entropic regularization of Sinkhorn divergences to the data's intrinsic geometry, and (iii) introduce a novel, intrinsic-dimension-based Richardson extrapolation estimator that strongly debiases Wasserstein distance estimation. Numerical experiments demonstrate that our geometry-aware pipeline effectively mitigates the discretization error bottleneck while maintaining computational efficiency.

Problem

• Optimal Transport
• sampling error
• intrinsic dimension
• Wasserstein distance
• discretization error

Innovation

• Geometry-Aware Optimal Transport
• Intrinsic Dimension Estimation
• Sampling Error Estimator
• Sinkhorn Divergence Calibration
• Richardson Extrapolation