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Limit laws of the empirical Wasserstein distance: Gaussian distributions [article]

Thomas Rippl, Axel Munk, Anja Sturm
2016 arXiv   pre-print
We derive central limit theorems for the Wasserstein distance between the empirical distributions of Gaussian samples.  ...  The cases are distinguished whether the underlying laws are the same or different. Results are based on the (quadratic) Frechet differentiability of the Wasserstein distance in the Gaussian case.  ...  Acknowledgement: Support of DFG RTN2088 is gratefully acknowledged. We are grateful to a referee, the associate editor and Michael Habeck for helpful comments.  ... 
arXiv:1507.04090v2 fatcat:rihugujrazgenj7shbuzuuurhu

Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples [article]

Yixing Zhang, Xiuyuan Cheng, Galen Reeves
2021 arXiv   pre-print
For the Gaussian-smoothed p-Wasserstein distance in d dimensions, our results require only the existence of a moment greater than d + 2p.  ...  of Gaussian smoothing.  ...  of optimal couplings converges in the σ → 0 limit to the optimal coupling for the unsmoothed Wasserstein distance.  ... 
arXiv:2103.00394v1 fatcat:o6hballeyncurpk4m3octmvwrm

Exact rate of convergence of the expected $W_{2}$ distance between the empirical and true Gaussian distribution

Philippe Berthet, Jean Claude Fort
2020 Electronic Journal of Probability  
We study the Wasserstein distance W_2 for Gaussian samples. We establish the exact rate of convergence √(loglog n/n) of the expected value of the W_2 distance between the empirical and true c.d.f.'  ...  s for the normal distribution. We also show that the rate of weak convergence is unexpectedly 1/√(n) in the case of two correlated Gaussian samples.  ...  Acknowledgements We are grateful to Michel Ledoux who pointed out the question of the exact limiting constant in (2) .  ... 
doi:10.1214/19-ejp410 fatcat:5zklbm5lgbh77pdd67ftv6fj4y

A Gaussian Process Regression Model for Distribution Inputs [article]

François Bachoc , Jean-Michel Loubes
2018 arXiv   pre-print
In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances.  ...  Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions  ...  We thank Yann Richet, from the French Radioprotection and Nuclear Safety Institute (IRSN), for introducing us to the problem of axial burn up analysis of fuel pins, which motivated the present work.  ... 
arXiv:1701.09055v2 fatcat:j7qow7g7enfythq5vigwcr32lm

Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding [article]

Francois Bachoc, Alexandra Suvorikova, David Ginsbourger, Jean-Michel Loubes, Vladimir Spokoiny
2019 arXiv   pre-print
by an empirical barycenter and additional explicit results in the special case of Gaussian distributions.  ...  While directly constructing radial positive definite kernels based on the Wasserstein distance has been proven to be possible in the unidimensional case, such constructions do not extend well to the multidimensional  ...  Clément Chevalier (now with the Swiss Statistical Office, Neuchâtel, Switzerland) who has been involved in investigations on this data set in the framework of the ReDICE consortium.  ... 
arXiv:1805.00753v2 fatcat:22cozcxfrvb6tfbs5jqddj5rqi

Distributional Gaussian Processes Layers for Out-of-Distribution Detection [article]

Sebastian G. Popescu, David J. Sharp, James H. Cole, Konstantinos Kamnitsas, Ben Glocker
2022 arXiv   pre-print
This directly replaces convolving Gaussian Processes with a distance-preserving affine operator on distributions.  ...  Gaussian Processes can reliably separate in-distribution data points from out-of-distribution data points via their mathematical construction.  ...  The Wasserstein-2 distance between two multivariate Gaussian distributions N (m 1 , Σ 1 ) and N (m 2 , Σ 2 ), which have associated Gaussian measures and implicitly the Wasserstein metric is well defined  ... 
arXiv:2206.13346v1 fatcat:lfgun4vokjgilkmxd7zc54iazq

A Gaussian Process Regression Model for Distribution Inputs

Francois Bachoc, Fabrice Gamboa, Jean-Michel Loubes, Nil Venet
2017 IEEE Transactions on Information Theory  
We thank Yann Richet, from the French Radioprotection and Nuclear Safety Institute (IRSN), for introducing us to the problem of axial burn up analysis of fuel pins, which motivated the present work.  ...  ACKNOWLEDGEMENTS We thank the anonymous reviewers, whose suggestions have greatly contributed to improve the manuscript.  ...  This approach is limited as the effect of such parameters do not take into account the whole shape of the law.  ... 
doi:10.1109/tit.2017.2762322 fatcat:7tc7zvti7vdc5nxpdjhl5kh6iu

Statistical Aspects of Wasserstein Distances

Victor M. Panaretos, Yoav Zemel
2018 Annual Review of Statistics and Its Application  
Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation.  ...  In this review, we provide a snapshot of the main concepts involved in Wasserstein distances and optimal transportation, and a succinct overview of some of their many statistical aspects.  ...  We thank a reviewer for comments on a preliminary version of the paper.  ... 
doi:10.1146/annurev-statistics-030718-104938 fatcat:5bmp6n5ohneavprhdjw2wop6qm

Kernel Density Estimation on Spaces of Gaussian Distributions and Symmetric Positive Definite Matrices

Emmanuel Chevallier, Emmanuel Kalunga, Jesús Angulo
2017 SIAM Journal of Imaging Sciences  
Explicit expressions of kernels are provided for the case of the 2-Wasserstein metric on multivariate Gaussian distributions and for the Fisher metric on multivariate centred distributions.  ...  This paper analyses the kernel density estimation on spaces of Gaussian distributions endowed with different metrics.  ...  has showed that the 2-Wasserstein distance on Gaussian laws is a Riemannian metric distance.  ... 
doi:10.1137/15m1053566 fatcat:6psum6wdbvectcl2buzydziwqa

Multivariate goodness-of-Fit tests based on Wasserstein distance [article]

Marc Hallin and Gilles Mordant and Johan Segers
2021 arXiv   pre-print
To this end, we prove a uniform law of large numbers for the empirical distribution in Wasserstein distance, where the uniformity is over any class of underlying distributions satisfying a uniform integrability  ...  The lack of asymptotic distribution theory for the empirical Wasserstein distance means that the validity of the parametric bootstrap under the null hypothesis remains a conjecture.  ...  Acknowledgements The authors gratefully acknowledge the remarks and comments made by the reviewers that greatly helped improve the paper. J.  ... 
arXiv:2003.06684v3 fatcat:etry3m46p5hx7hivn755ntka7e

Inference for Empirical Wasserstein Distances on Finite Spaces [article]

Max Sommerfeld, Axel Munk
2017 arXiv   pre-print
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 program  ...  Tameling for careful reading of the manuscript.  ... 
arXiv:1610.03287v2 fatcat:fjaiohyijffupnin6tvhebfgzi

Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance [article]

Xianliang Xu, Zhongyi Huang
2022 arXiv   pre-print
We utilize the central limit theorem in Banach space to derive the limit distribution for the Sliced 1-Wasserstein distance.  ...  But due to its high computational complexity and the phenomenon of the curse of dimensionality in empirical estimation, various extensions of the Wasserstein distance have been proposed to overcome the  ...  Central Limit Theorem for the Sliced 1-Wasserstein Distance In this section, we explore the limit distribution of Sliced 1-Wasserstein distance between the empirical and the true probability measure.  ... 
arXiv:2205.14624v2 fatcat:5izo7f7ex5bv7ehm3vb3ikfppy

Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications [article]

Ritwik Sadhu and Ziv Goldfeld and Kengo Kato
2022 arXiv   pre-print
The smooth 1-Wasserstein distance (SWD) W_1^σ was recently proposed as a means to mitigate the curse of dimensionality in empirical approximation while preserving the Wasserstein structure.  ...  As applications of the limit distribution theory, we study two-sample testing and minimum distance estimation (MDE) under W_1^σ.  ...  Goldfeld is supported by the NSF CRII Grant CCF-1947801, the NSF CAREER Award under Grant CCF-2046018, and the 2020 IBM Academic Award. K.  ... 
arXiv:2107.13494v5 fatcat:pvywb6rdxnbpnatys3hvsial3u

Schoenberg-Rao distances: Entropy-based and geometry-aware statistical Hilbert distances [article]

Gaëtan Hadjeres, Frank Nielsen
2020 arXiv   pre-print
Distances between probability distributions that take into account the geometry of their sample space,like the Wasserstein or the Maximum Mean Discrepancy (MMD) distances have received a lot of attention  ...  In particular, we introduce a principled way to construct such kernels and derive novel closed-form distances between mixtures of Gaussian distributions.  ...  Through the discussion of the related work and experiments, we highlighted the connections and differences between MMD [17] , GAIT [14] and Wasserstein [10] distances.  ... 
arXiv:2002.08345v2 fatcat:x64srmc62jfkhmlyd6aszfli2e

Multivariate goodness-of-fit tests based on Wasserstein distance

Marc Hallin, Gilles Mordant, Johan Segers
2021 Electronic Journal of Statistics  
To this end, we prove a uniform law of large numbers for the empirical distribution in Wasserstein distance, where the uniformity is over any class of underlying distributions satisfying a uniform integrability  ...  The lack of asymptotic distribution theory for the empirical Wasserstein distance means that the validity of the parametric bootstrap under the null hypothesis remains a conjecture.  ...  Acknowledgments The authors gratefully acknowledge the remarks and comments made by the reviewers that greatly helped improve the paper.  ... 
doi:10.1214/21-ejs1816 fatcat:gu4q6kvk2vable6ohl6uurgsgm
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