Journal ArticleDOI
The Wasserstein distance and approximation theorems
TLDR
An explicit formula for the Wasserstein distance between multivariate distributions in certain cases is obtained by an extension of the idea of the multivariate quantile transform and some applications are given to the problem of approximation of stochastic processes by simpler ones.Abstract:
By an extension of the idea of the multivariate quantile transform we obtain an explicit formula for the Wasserstein distance between multivariate distributions in certain cases For the general case we use a modification of the definition of the Wasserstein distance and determine optimal ‘markov-constructions’ We give some applications to the problem of approximation of stochastic processes by simpler ones, as eg weakly dependent processes by independent sequences and, finally, determine the optimal martingale approximation to a given sequence of random variables; the Doob decomposition gives only the ‘one-step optimal’ approximationread more
Citations
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Book
Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Book ChapterDOI
Toward a Systematic Approach to the Design and Evaluation of Automated Mobility-on-Demand Systems: A Case Study in Singapore
TL;DR: Using actual transportation data, this analysis suggests a shared-vehicle mobility solution can meet the personal mobility needs of the entire population with a fleet whose size is approximately 1/3 of the total number of passenger vehicles currently in operation.
Book
One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances
Sergey G. Bobkov,Michel Ledoux +1 more
TL;DR: In this paper, the authors studied the convergence of the empirical measures μn = 1 n ∑n k=1 δXk, n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp.
Journal ArticleDOI
A characterization of random variables with minimum L 2 -distance
TL;DR: In this article, a complete characterization of multivariate random variables with minimum L 2 Wasserstein distance is proved by means of duality theory and convex analysis, which allows to determine explicitly the optimal couplings for several multivariate distributions.
Journal ArticleDOI
On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces
TL;DR: For a separable metric space (X, d) Lp Wasserstein metrics between probability measures μ and v on X are defined by as mentioned in this paper where the infimum is taken over all probability measures η on X × X with marginal distributions ρ and v, respectively.
References
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Book
Inequalities: Theory of Majorization and Its Applications
TL;DR: In this paper, Doubly Stochastic Matrices and Schur-Convex Functions are used to represent matrix functions in the context of matrix factorizations, compounds, direct products and M-matrices.
Book
Convex analysis and measurable multifunctions
Charles Castaing,Michel Valadier +1 more
TL;DR: In this paper, the authors consider convex functions with topological properties of the profile of a convex multifunction with compact convex values and prove the compactness theorems of measurable selections and integral representation theorem.
Journal ArticleDOI
The Fréchet distance between multivariate normal distributions
D.C Dowson,B.V Landau +1 more
TL;DR: The Frechet distance between two multivariate normal distributions having means μX, μY and covariance matrices ΣX, ΣY is shown to be given by d2 = |μX − μY|2 + tr(ΣX + ǫ − 2(ǫ) 12) as discussed by the authors.