DeepAI AI Chat
Log In Sign Up

On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure

by   Louis Brown, et al.
Yale University

We discuss the classical problem of measuring the regularity of distribution of sets of N points in T^d. A recent line of investigation is to study the cost (= mass × distance) necessary to move Dirac measures placed in these points to the uniform distribution. We show that Kronecker sequences satisfy optimal transport distance in d ≥ 3 dimensions. This shows that for differentiable f: T^d →R and badly approximable vectors α∈R^d, we have | ∫_T^d f(x) dx - 1/N∑_k=1^N f(k α) | ≤ c_α∇ f^(d-1)/d_L^∞∇ f^1/d_L^2/N^1/d. We note that the result is uniformly true for a sequence instead of a set. Simultaneously, it refines the classical integration error for Lipschitz functions, ∇ f_L^∞ N^-1/d. We obtain a similar improvement for numerical integration with respect to the regular grid. The main ingredient is an estimate involving Fourier coefficients of a measure; this allows for existing estimates to be conviently `recycled'. We present several open problems.


page 1

page 2

page 3

page 4


Bounding quantiles of Wasserstein distance between true and empirical measure

Consider the empirical measure, P̂_N, associated to N i.i.d. samples of ...

Transport type metrics on the space of probability measures involving singular base measures

We develop the theory of a metric, which we call the ν-based Wasserstein...

Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

We revisit the question of characterizing the convergence rate of plug-i...

Sequences of well-distributed vertices on graphs and spectral bounds on optimal transport

Given a graph G=(V,E), suppose we are interested in selecting a sequence...

Geophysical Inversion and Optimal Transport

We propose a new approach to measuring the agreement between two oscilla...

Transport Dependency: Optimal Transport Based Dependency Measures

Finding meaningful ways to determine the dependency between two random v...

Estimating processes in adapted Wasserstein distance

A number of researchers have independently introduced topologies on the ...