One-dimensional empirical measures, order
                    statistics, and Kantorovich transport
                    distances

Bobkov, Sergey G.; Ledoux, Michel

doi:10.1090/memo/1259

Cited by 163 publications

(246 citation statements)

References 80 publications

Supporting

Mentioning

241

Contrasting

Order By: Relevance

“…Let now µ be the standard Gaussian measure on the Borel sets of R d . It has been shown in [5] that in dimension d = 1, With respect to (1.3), it therefore appears that, already in dimension one, the rates for p ≥ 2 are rather sensitive to the underlying distribution. The proof of (1.4) in [5] relies on monotone rearrangement transport and explicit one-dimensional distributional inequalities.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Optimal Matching of Gaussian Samples

Ledoux

2019

J Math Sci

View full text Add to dashboard Cite

This article is a continuation of the papers [8,9] in which the optimal matching problem, and the related rates of convergence of empirical measures for Gaussian samples are addressed. A further step in both the dimensional and Kantorovich parameters is achieved here, proving that, given X 1 , . . . , X n independent random variables with common distribution the standard Gaussian measure µ on R d , d ≥ 3, and µ n = 1 n n i=1 δ X i the associated empirical measure, E W p p (µ n , µ) ≈

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Optimal Matching of Gaussian Samples

Ledoux

2019

J Math Sci

View full text Add to dashboard Cite

show abstract

“…Distributional limits give a natural perspective for practicable inference but despite considerable interest in the topic have remained elusive to a large extent. For measures on X = R quite a complete theory is available (see Munk and Czado (1998), Freitag et al (2007) and Freitag and Munk (2005) for μ 0 = μ and for example Del Barrio et al (1999), Samworth and Johnson (2005) and Del Barrio et al (2005) for μ 0 = μ as well as Mason (2016) and Bobkov and Ledoux (2014) for recent surveys). However, for X = R d , d 2, the only distributional result known to us is due to Rippl et al (2015) for specific multivariate (elliptic) parametric classes of distributions, when the empirical measure is replaced by a parametric estimate.…”

Section: Related Work 141 Empirical Wasserstein Distancesmentioning

confidence: 99%

Inference for Empirical Wasserstein Distances on Finite Spaces

Sommerfeld

Munk²

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

149

142

View full text Add to dashboard Cite

Summary. The Wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits. To overcome this obstacle, for probability measures supported on finitely many points, we derive the asymptotic distribution of empirical Wasserstein distances as the optimal value of a linear programme with random objective function.This facilitates statistical inference (e.g. confidence intervals for sample-based Wasserstein distances) in large generality. Our proof is based on directional Hadamard differentiability. Failure of the classical bootstrap and alternatives are discussed. The utility of the distributional results is illustrated on two data sets.

show abstract

“…If, at size N , the points are sampled uniformly in the domain Ω N := N 1 d Ω (as to keep the average density of order 1), we just have to scale the result above by the factor N p d . Recently also the case in which Ω is not compact, and the points are not sampled uniformly, has been considered [17][18][19].…”

mentioning

confidence: 99%

The Dyck bound in the concave 1-dimensional random assignment model

Caracciolo¹,

D'Achille²,

Erba³

et al. 2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We consider models of assignment for random N blue points and N red points on an interval of length 2N , in which the cost for connecting a blue point in x to a red point in y is the concave function |x − y| p , for 0 < p < 1. Contrarily to the convex case p > 1, where the optimal matching is trivially determined, here the optimization is non-trivial.The purpose of this paper is to introduce a special configuration, that we call the Dyck matching, and to study its statistical properties. We compute exactly the average cost, in the asymptotic limit of large N , together with the first subleading correction. The scaling is remarkable: it is of order N for p < 1 2 , order N ln N for p = 1 2 , and N arXiv:1904.10867v2 [cond-mat.dis-nn]In this paper we study the statistical properties of the Euclidean random assignment problem, in the case in which the points are confined to a one-dimensional interval, and the cost is a concave increasing function of their distance. The assignment problem is a combinatorial optimization problem, a special case of the matching problem when the underlying graph is bipartite. As for any combinatorial optimization problem, each realisation of the problem is described by an instance J, and the goal is to find, within some space of configurations, the particular one that minimises the given cost function. In the assignment problem, the instance J is a real-positive N × N matrix, encoding the costs of each possible pairing among N blue and N red points (J ij is the cost of pairing the i-th blue point with the j-th red point), the space of configurations is the set of permutations of N objects, π ∈ S N (describing a complete assignment of blue and red points), and the cost function isA random assignment problem is the datum of a probability measure µ N (J) on the possible instances of the problem. The interest is in the determination of the statistical properties of this problem w.r.t. the measure under analysis, and in particular the statistical properties of the optimal configuration. The problem can be formulated as the zero-temperature limit of the statistical mechanics properties of a disordered system, where the disorder is the instance J, the dynamical variables are encoded by π, and the Hamiltonian is the cost function H J (π).The case of random assignment problem in which the entries J ij are random i.i.d. variables, presented already in [1], has been solved at first, in a seminal paper by Parisi and Mézard [2], through the replica trick and afterwards by the Cavity Equations [3] (see also [4] for a recent generalization of those results also at finite system size). The Parisi-Mézard solution also leads to the striking prediction that, calling π opt (J) the optimal matching for the instance J and H opt (J) = H J (π opt (J)) its optimal cost, the average over all instances of H opt , for N large, tends to π 2 6 . This problem is simpler than a spin glass, as the determination of the optimal configuration is feasible in polynomial time (for example, through the celebrated Hungarian Alg...

show abstract

One-dimensional empirical measures, order statistics, and Kantorovich transport distances

Cited by 163 publications

References 80 publications

On Optimal Matching of Gaussian Samples

On Optimal Matching of Gaussian Samples

Inference for Empirical Wasserstein Distances on Finite Spaces

The Dyck bound in the concave 1-dimensional random assignment model

Contact Info

Product

Resources

About