[Paper Review] On the rate of convergence of empirical measures in $\infty$-transportation distance

Nicolás García Trillos, Dejan Slepčev|arXiv (Cornell University)|Jul 4, 2014

Geometry and complex manifolds9 references3 citations

TL;DR

This paper establishes an optimal upper bound on the rate of convergence of empirical measures in the ∞-transportation distance for i.i.d. samples from an absolutely continuous probability measure on a bounded, connected domain with Lipschitz boundary. The bound scales as $ \frac{\ln(n)^{3/4}}{n^{1/2}} $ in 2D and $ \frac{\ln(n)^{1/d}}{n^{1/d}} $ in $ d \geq 3 $, with high probability, and is shown to be tight in terms of scaling with sample size $ n $.

ABSTRACT

We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty$-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.

Motivation & Objective

• Address the lack of sharp convergence rate estimates for the ∞-transportation distance between a probability measure and its empirical counterpart.
• Establish a non-asymptotic upper bound on the ∞-transportation distance that is optimal in scaling with the number of samples $ n $.
• Extend prior results on min-max matching and optimal transport to general absolutely continuous measures with bounded density on domains with Lipschitz boundaries.
• Provide a probabilistic analysis of the convergence rate under minimal regularity assumptions on the underlying measure.
• Confirm the optimality of the derived bounds in terms of $ n $-scaling, particularly highlighting the logarithmic corrections in the 2D case.

Proposed method

• Use a dyadic decomposition of the domain into dyadic cubes to control local discrepancies between the empirical and true measures.
• Construct bi-Lipschitz mappings between subregions of the domain and reference cubes to transfer matching estimates from regular grids to irregular domains.
• Apply concentration inequalities and union bounds over dyadic levels to control the maximal displacement across all sample points.
• Utilize the fact that the density is uniformly bounded away from zero and infinity to ensure uniform sampling behavior across the domain.
• Establish that the ∞-transportation distance is equivalent to the min-max matching distance in the discrete setting, enabling the use of matching theory techniques.
• Prove that the derived bound is optimal by comparing it to known lower bounds in the literature for the Lebesgue measure on the unit cube.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence of the ∞-transportation distance between a probability measure and its empirical measure in terms of the number of samples $ n $?
• RQ2 How does the convergence rate depend on the dimension $ d $, particularly in the 2D and $ d \geq 3 $ cases?
• RQ3 Can the convergence rate be improved or tightened when the underlying measure has a bounded density on a domain with Lipschitz boundary?
• RQ4 Is the logarithmic correction factor $ \ln(n)^{3/4} $ in 2D and $ \ln(n)^{1/d} $ in higher dimensions necessary and optimal?
• RQ5 Does the bound hold with high probability, and can it be shown to be tight in terms of $ n $-scaling?

Key findings

• The paper establishes an upper bound on the ∞-transportation distance between the true measure $ \nu $ and the empirical measure $ \nu_n $ that scales as $ \frac{\ln(n)^{3/4}}{n^{1/2}} $ in two dimensions.
• In dimensions $ d \geq 3 $, the upper bound scales as $ \frac{\ln(n)^{1/d}}{n^{1/d}} $, which matches known lower bounds up to logarithmic factors.
• The bound holds with high probability, specifically except on a set of probability $ O(n^{-\alpha/2}) $ for any fixed $ \alpha > 2 $.
• The derived bound is optimal in terms of scaling with $ n $, meaning no faster convergence rate is possible under the given assumptions.
• The result extends prior work on min-max matching in the unit cube to general absolutely continuous measures with bounded densities on domains with Lipschitz boundaries.
• The proof relies on a domain decomposition into dyadic cubes and the construction of bi-Lipschitz mappings to transfer matching estimates from regular grids to the domain’s geometry.