An Invitation to Statistics in Wasserstein Space

Panaretos, Victor M.; Zemel, Yoav

doi:10.1007/978-3-030-38438-8

Cited by 93 publications

(76 citation statements)

References 101 publications

(198 reference statements)

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“…More is known about the topology of Wasserstein space; for instance, the exponential and log maps given in Section 3 are continuous, so WpHq is homeomorphic to an infinite-dimensional convex subset of a Hilbert space L 2 pµq (for any regular measure µ); see the dissertation Zemel [59, Lemmas 3.4.4 and 3.4.5] or the forthcoming book Panaretos and Zemel [43].…”

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confidence: 99%

Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

2018

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Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinitedimensionality of these operators. In this paper, we describe the manifold geometry of the space of trace-class infinite-dimensional covariance operators and associated key statistical properties, under the recently proposed infinite-dimensional version of the Procrustes metric. We identify this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation. The identification allows us to provide a complete description of those aspects of this manifold geometry that are important in terms of statistical inference, and establish key properties of the Fréchet mean of a random sample of covariances, as well as generative models that are canonical for such metrics and link with the problem of registration of functional data.

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confidence: 99%

Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

2018

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“…The question thus arises to what extent the measures P S , P U , P X are compatible in the sense of composition of optimal transportation maps, as in Definition 2.3.1 in Panaretos and Zemel [15], i.e., to determine whether the proposed copula-marginal two steps approach is equivalent to the direct approach of Chernozhukov et al [3].…”

Section: Discussionmentioning

confidence: 99%

“…In the continuously differentiable case, Panaretos and Zemel [15] p. 50 show that a necessary condition is the commutativity of gradient of the transport maps. In our setting, this corresponds to the commutativity of the matrices ∇G −1 (Q C (s))) and ∇Q C (s).…”

Section: Discussionmentioning

confidence: 99%

“…i) The proof is based on the aforementioned general result of uniform convergence on compacta of optimal transportation maps of Panaretos and Zemel [15], Proposition 1.7.11. The latter does not require a convexity assumption on the support of the measures, nor a homeomophism condition of the transport maps, as mandated by previous results in the literature, such as Theorems A.1 and A.2 in Chernozhukov et al [3] (see also Theorem 1.7.7 in Panaretos and Zemel [15]). The second main ingredient of the proof is the continuity result for optimal mappings of Cuesta-Albertos et al [4].…”

Section: Remarkmentioning

confidence: 99%

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Functional, randomized and smoothed multivariate quantile regions

Faugeras

Rüschendorf

2021

Journal of Multivariate Analysis

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The mass transportation approach to multivariate quantiles in Chernozhukov et al. [3] was modified in Faugeras and Rüschendorf [8] by a two steps procedure. In the first step, a mass transportation problem from a spherical reference measure to the copula is solved and combined in the second step with a marginal quantile transformation in the sample space. Also, generalized quantiles given by suitable Markov morphisms are introduced there.In the present paper, this approach is further extended by a functional approach in terms of membership functions, and by the introduction of randomized quantile regions. In addition, in the case of continuous marginals, a smoothed version of the empirical quantile regions is obtained by smoothing the empirical copula. All three extended approaches give empirical quantile ares of exact level and improved stability. The resulting depth areas give a valid representation of the central quantile areas of a multivariate distribution and provide a valuable tool for their analysis.

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“…As such, Wasserstein distances broaden the scope of optimal transport theory to probability theory. Additionally, they have been exploited to move further beyond these realms to solve concrete problems in inferential statistics, such as in Panaretos and Zemel [39]. Establishing Wasserstein distances in tropical geometric settings thus provides a framework for a vast body of existing results in these related fields to be applicable to the important problem of statistical inference and data analysis in applied tropical geometric settings by providing a setting for the study of probability measures and distributions.…”

Section: Introductionmentioning

confidence: 99%

Tropical optimal transport and Wasserstein distances

Lee

Lin

et al. 2021

Info. Geo.

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We study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees—as the ground metric and study the cases of $$p=1,2$$ p = 1 , 2 in detail. The case of $$p=1$$ p = 1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $$p=2$$ p = 2 gives a form for Fréchet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.

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An Invitation to Statistics in Wasserstein Space

Cited by 93 publications

References 101 publications

Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

Functional, randomized and smoothed multivariate quantile regions

Tropical optimal transport and Wasserstein distances

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