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

TL;DR

This paper introduces geometry-aware estimators for optimal transport that accurately measure sampling error and intrinsic dimension, enabling faster and more reliable Wasserstein distance estimation in high-dimensional data.

Contribution

The work presents novel, tuning-free estimators for sampling error and intrinsic dimension that improve the accuracy and efficiency of optimal transport computations.

Findings

A tuning-free estimator for OT cost that requires no solver.

A fast intrinsic dimension estimator from multi-scale decay.

Enhanced Wasserstein distance estimation with reduced discretization error.

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.