[Paper Review] A short proof on the rate of convergence of the empirical measure for the Wasserstein distance

Vincent Divol|arXiv (Cornell University)|Jan 20, 2021

Geometric Analysis and Curvature Flows5 references4 citations

TL;DR

This paper provides a concise proof that the expected Wasserstein distance between the empirical measure and the true measure on the d-dimensional flat torus converges at rate $ n^{-1/d} $ when the true measure has a bounded density. The approach leverages a connection between Wasserstein distance and negative Sobolev norms, combined with Fourier analysis and kernel smoothing, to establish tight convergence rates that match minimax lower bounds up to logarithmic factors in dimension 2.

ABSTRACT

We provide a short proof that the Wasserstein distance between the empirical measure of a n-sample and the estimated measure is of order n^-(1/d), if the measure has a lower and upper bounded density on the d-dimensional flat torus.

Motivation & Objective

• To establish a tight, short proof of the convergence rate of the empirical measure to the true measure in the Wasserstein distance under bounded density assumptions.
• To avoid boundary effects by restricting the domain to the d-dimensional flat torus instead of a convex domain.
• To demonstrate that the $ n^{-1/d} $ rate is optimal (up to logarithmic factors in dimension 2) using tools from Fourier analysis and optimal transport.
• To unify and simplify existing approaches by connecting Wasserstein distance to negative Sobolev norms via a key inequality from optimal transport theory.

Proposed method

• Use of the inequality $ W_p(\mu, \nu) \leq p f_{\min}^{1/p - 1} \|f - g\|_{\dot{H}_p^{-1}} $, which links Wasserstein distance to the negative Sobolev norm of the density difference.
• Application of kernel smoothing to regularize the empirical measure, with $ \mu_{n,h} $ denoting the smoothed empirical measure via convolution with a smooth kernel $ K_h $.
• Bounding the Wasserstein distance between $ \mu_n $ and $ \mu_{n,h} $ by $ C_0 h $, using the cost of transporting mass through the kernel convolution.
• Expressing the negative Sobolev norm via Fourier multipliers, specifically using the operator $ \mathcal{A} $ defined by $ a(\xi) = 1/|\xi| $ for $ |\xi| \geq 1 $.
• Applying the Mikhlin multiplier theorem to control the $ L_p $-norm of $ \mathcal{A}(f_{n,h} - f_h) $, ensuring boundedness of the associated operator.
• Using Rosenthal's inequality to control the $ L_p $-norm of the sum of i.i.d. centered random functions $ \mathcal{A}(f_{n,h} - f_h) $, decomposing the error into bias and fluctuation terms.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence of the empirical measure to the true measure in the Wasserstein distance under bounded density assumptions?
• RQ2 Can the $ n^{-1/d} $ convergence rate be proven concisely using Fourier analysis and negative Sobolev norms?
• RQ3 How does the convergence rate depend on the dimension $ d $, especially in low dimensions ($ d=1,2 $)?
• RQ4 Is the $ n^{-1/d} $ rate minimax optimal for the class of measures with bounded densities on the flat torus?

Key findings

• The expected $ W_p $-distance between the empirical measure $ \mu_n $ and the true measure $ \mu $ is bounded by $ C n^{-1/d} $ for $ d \geq 3 $, matching the minimax rate up to constants.
• For $ d = 2 $, the rate is $ (\log n)^{1/2} n^{-1/2} $, which matches the minimax rate up to a logarithmic factor.
• For $ d = 1 $, the rate is $ n^{-1/2} $, consistent with the known minimax optimal rate.
• The proof establishes that the $ n^{-1/d} $ rate is tight for $ d \geq 3 $, and the result is optimal up to logarithmic factors in dimension 2.
• The method avoids dyadic partitioning and instead uses a Fourier-analytic approach via negative Sobolev norms, simplifying the analysis.
• The same framework can be extended to show that with $ s $-Hölder smooth densities, the rate improves to $ n^{-(s+1)/(2s + d)} $, matching known minimax results.