Distribution’s template estimate with Wasserstein metrics

Boissard, Emmanuel; Gouic, Thibaut Le; Loubes, Jean–Michel

doi:10.3150/13-bej585

Cited by 75 publications

(101 citation statements)

References 26 publications

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“…The barycentre with respect to the Wasserstein metric (Agueh and Carlier, 2011) has been shown to elicit important structure from complex data and to be a promising tool, e.g. in deformable models (Boissard et al, 2015;Agulló-Antolín et al, 2015). It has also been used in large-scale Bayesian inference to combine posterior distributions from subsets of the data (Srivastava et al, 2015).…”

Section: Introductionmentioning

confidence: 99%

Inference for Empirical Wasserstein Distances on Finite Spaces

Sommerfeld

Munk²

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

145

142

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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.

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Section: Introductionmentioning

confidence: 99%

Inference for Empirical Wasserstein Distances on Finite Spaces

Sommerfeld

Munk²

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

145

142

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“…However, there is a metric structure compatible with this topology called Wasserstein distance, which arises from the field of optimal transport (see, e.g., Santambrogio, 2015;Villani, 2003). The Wasserstein distance has attracted a lot of interest recently and has gained a lot of popularity in various fields (e.g., Boissard, Le Gouic, & Loubes, 2015;Burger, Franek, & Schönlieb, 2012;Irpino & Verde, 2008;Ni, Bresson, Chan, & Esedoglu, 2009;Piccoli & Rossi, 2014).…”

Section: The Wasserstein Barycentermentioning

confidence: 99%

Model aggregation using optimal transport and applications in wind speed forecasting

2018

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In several environmental affected socio‐economic activities, including renewable energy site assessment, search and rescue operations, and local microclimate modeling, the need of very local wind speed prediction is critical and not completely covered by the use of numerical weather prediction models. In meteorology and, particularly, in wind speed prediction, the spatial location of the prediction does not coincide with the spatial locations where numerical models provide estimates of the relevant quantity (which are typically grid points used for the numerical resolution of the wind transport equations). Hence, the important problem of constructing a predictive model for the wind speed at the required location using a combination of actual measurements and model predictions arises. This problem is far from trivial on account of the fact that measurements and predictions do not refer to the same quantity for the reason that typical grid points for the numerical scheme that provide model predictions and the location of the meteorological stations that provide measurements do not coincide. In this work, a new approach is proposed based on optimal transportation theory for the aggregation of model predictions and measurements for the construction of an optimal predictor for wind speed at the location of interest. Our model provides a linear predictive model in the space of probability distributions of the predictors (Wasserstein space), which is then mapped into observation space using a generalized quantile regression technique. Importantly, the proposed scheme allows also for the construction of zone monitoring the extremes, which when applied to real data, provides superior results with respect to other existing methods.

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“…Given a complete and separable metric space X, one defines its Wasserstein space as the collection of sufficiently concentrated Borel probability measures endowed with a metric which is calculated by means of optimal transport (see the precise definitions in Subsection 1.3). This notion has strong connections to many flourishing areas in pure and applied mathematics including probability theory [5,6], theory of (stochastic) partial differential equations [11,12], geometry of metric spaces [17,20,22], machine learning [2,21], and many more. Besides of these connections, the p-Wasserstein space itself is an interesting object, being a measure theoretic analogue of L p spaces [14].…”

mentioning

confidence: 99%

Isometric study of Wasserstein spaces – the real line

Gehér¹,

Titkos

Virosztek

2020

Trans. Amer. Math. Soc.

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Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W2(R n ). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom (Wp(R)), the isometry group of the Wasserstein space Wp(R) for all p ∈ [1, ∞) \ {2}. We show that W2(R) is also exceptional regarding the parameter p: Wp(R) is isometrically rigid if and only if p = 2. Regarding the underlying space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact interval [0,1]. Surprisingly, in that case, Wp([0, 1]) is isometrically rigid if and only if p = 1. Moreover, W1([0, 1]) admits isometries that split mass, and Isom (W1([0, 1])) cannot be embedded into Isom (W1(R)).

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Distribution’s template estimate with Wasserstein metrics

Cited by 75 publications

References 26 publications

Inference for Empirical Wasserstein Distances on Finite Spaces

Inference for Empirical Wasserstein Distances on Finite Spaces

Model aggregation using optimal transport and applications in wind speed forecasting

Isometric study of Wasserstein spaces – the real line

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