[Paper Review] A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation

Adrien Vacher, Boris Muzellec+3 more|arXiv (Cornell University)|Jan 13, 2021

Markov Chains and Monte Carlo Methods44 references9 citations

TL;DR

This paper proposes a novel computational estimator for smooth optimal transport that achieves dimension-free statistical and computational rates by leveraging an infinite-dimensional sum-of-squares representation. For smooth distributions with regularity parameter $ m > d $, the method requires $ \tilde{O}(\varepsilon^{-2}) $ samples and $ \tilde{O}(\varepsilon^{-4}) $ computation time when $ m \geq 3d $, yielding dimension-independent complexity despite dimension-dependent constants, thus closing the statistical-computational gap in smooth OT estimation.

ABSTRACT

It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimensionality. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexity of these recently proposed methods still degrades exponentially with the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision $\varepsilon$ from $ ilde{O}(\varepsilon^{-2})$ independent and identically distributed samples from the distributions, for a computational cost of $ ilde{O}(\varepsilon^{-4})$ when the smoothness increases, hence yielding dimension-free statistical and computational rates, with potentially exponentially dimension-dependent constants.

Motivation & Objective

• To address the statistical-computational gap in optimal transport estimation under smoothness assumptions.
• To develop a tractable, polynomial-time algorithm that matches the minimax statistical rates for smooth distributions.
• To leverage smoothness not only statistically but also computationally, overcoming the curse of dimensionality in existing methods.
• To unify estimation under i.i.d. sampling, exact integration, and pointwise density evaluation via kernel mean embeddings.

Proposed method

• The method uses an infinite-dimensional sum-of-squares (SoS) representation of the optimal transport potential to enable efficient computation.
• It formulates the dual optimal transport problem as a regularized semidefinite program (SDP) with kernel-based constraints and marginal penalty terms.
• The estimator is constructed using kernel mean embeddings to represent measures, enabling unified analysis across sampling, integration, and evaluation scenarios.
• A primal-dual algorithm with regularization parameters $ \lambda_1 $ and $ \lambda_2 $ is used to solve the SDP, with convergence guarantees via strong duality.
• The method introduces a space-filling sampling strategy using quasi-random sequences to approximate the dual constraints efficiently.
• The final estimator $ \hat{d}_{\text{OT}} $ is derived from the dual potentials $ \hat{u}, \hat{v} $, with the transportation map inferred as $ \hat{T}(x) = x - \nabla_x \hat{u}(x) $.

Experimental results

Research questions

• RQ1 Can smoothness of the optimal transport potentials be exploited computationally to achieve dimension-free complexity?
• RQ2 Is there a tractable, polynomial-time algorithm that matches the minimax statistical rates for smooth optimal transport?
• RQ3 Can a unified framework be developed for i.i.d. sampling, exact integration, and pointwise density evaluation in OT estimation?
• RQ4 What is the computational cost of achieving $ \varepsilon $-accuracy in smooth OT estimation, and how does it scale with dimension and smoothness?

Key findings

• The proposed estimator achieves $ \tilde{O}(\varepsilon^{-2}) $ sample complexity and $ \tilde{O}(\varepsilon^{-4}) $ computational cost when $ m \geq 3d $, making the complexity dimension-free in the exponent.
• For $ m < 3d $, the computational cost scales as $ \tilde{O}(\varepsilon^{-\max(4, \frac{7d}{m} - d)}) $, improving with increasing smoothness $ m $.
• When exact integrals or pointwise density evaluations are available, the computational cost can be smaller than $ \varepsilon^{-4} $, reducing reliance on Monte Carlo sampling.
• Numerical experiments confirm convergence of the estimator $ \hat{d}_{\text{OT}} $ to the true OT distance on 4D truncated Gaussians, with error decreasing as sample size increases.
• The method recovers the unregularized OT problem in the limit $ \lambda_1, \lambda_2 \to 0 $, validating its consistency with classical OT.
• Theoretical analysis shows the total error is bounded by $ O(\varepsilon) $, with explicit dependence on smoothness, regularization, and approximation error.