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

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 $\text{OT}_c(ρ, \hatρ)$ 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.

Decreasing Entropic Regularization Averaged Gradient for Semi-Discrete Optimal Transport

Adding entropic regularization to Optimal Transport (OT) problems has become a standard approach for designing efficient and scalable solvers. However, regularization introduces a bias from the true solution. To mitigate this bias while still benefiting from the acceleration provided by regularization, a natural solver would adaptively decrease the regularization as it approaches the solution. Although some algorithms heuristically implement this idea, their theoretical guarantees and the extent of their acceleration compared to using a fixed regularization remain largely open. In the setting of semi-discrete OT, where the source measure is continuous and the target is discrete, we prove that decreasing the regularization can indeed accelerate convergence. To this end, we introduce DRAG: Decreasing (entropic) Regularization Averaged Gradient, a stochastic gradient descent algorithm where the regularization decreases with the number of optimization steps. We provide a theoretical analysis showing that DRAG benefits from decreasing regularization compared to a fixed scheme, achieving an unbiased $\mathcal{O}(1/t)$ sample and iteration complexity for both the OT cost and the potential estimation, and a $\mathcal{O}(1/\sqrt{t})$ rate for the OT map. Our theoretical findings are supported by numerical experiments that validate the effectiveness of DRAG and highlight its practical advantages.