On Wasserstein geometry of Gaussian measures

Takatsu, Asuka

doi:10.2969/aspm/05710463

Cited by 115 publications

(163 citation statements)

References 10 publications

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“…The µ i 's will be compatible if the covariances commute, in which case we have the explicit solution Σ 1/2 = n −1 (Σ 1/2 1 + · · · + Σ 1/2 n ), but otherwise there is no explicit expression for the Fréchet mean. The restriction of W 2 (R d ) to Gaussian measures leads to a stratified space, whose geometry was studied carefully by Takatsu (2011), including expressions for the curvature. In particular, the latter grows without bound as one approaches singular covariance matrices.…”

Section: Gaussian Measuresmentioning

confidence: 99%

Statistical Aspects of Wasserstein Distances

Panaretos

Zemel

2019

Annu. Rev. Stat. Appl.

416

173

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Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in order to recover the other distribution. They are ubiquitous in mathematics, with a long history that has seen them catalyse core developments in analysis, optimization, and probability. Beyond their intrinsic mathematical richness, they possess attractive features that make them a versatile tool for the statistician: they can be used to derive weak convergence and convergence of moments, and can be easily bounded; they are well-adapted to quantify a natural notion of perturbation of a probability distribution; and they seamlessly incorporate the geometry of the domain of the distributions in question, thus being useful for contrasting complex objects. Consequently, they frequently appear in the development of statistical theory and inferential methodology, and have recently become an object of inference in themselves. 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.

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Section: Gaussian Measuresmentioning

confidence: 99%

Statistical Aspects of Wasserstein Distances

Panaretos

Zemel

2019

Annu. Rev. Stat. Appl.

416

173

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“…(1. 10)]. From the large deviation results of now classic papers, 3, 4, 6 the time-discrete one used here can be derived, e.g., using the contraction principle.…”

Section: Introductionmentioning

confidence: 99%

Upscaling from particle models to entropic gradient flows

Dirr

Laschos

Zimmer

2012

Journal of Mathematical Physics

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We prove that, for the case of Gaussians on the real line, the functional derived by a time discretization of the diffusion equation as entropic gradient flow is asymptotically equivalent to the rate functional derived from the underlying microscopic process. This result strengthens a conjecture that the same statement is actually true for all measures with second finite moment. C 2012 American Institute of Physics.

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“…It plays an important role in both these subjects. We refer the reader to [18] for a recent exposition, and to [12,26,29,37] for earlier work. The quantity F (A, B) = tr(A 1/2 BA 1/2 ) 1/2 is called the fidelity between the states A and B.…”

Section: Introductionmentioning

confidence: 99%

Matrix versions of the Hellinger distance

2019

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On the space of positive definite matrices we consider distance functions of the form d(A, B) = [trA(A, B) − trG(A, B)]1/2 , where A(A, B) is the arithmetic mean and G(A, B) is one of the different versions of the geometric mean. When G(A, B) = A 1/2 B 1/2 this distance is A 1/2 − B 1/2 2 , and when G(A, B) = (A 1/2 BA 1/2 ) 1/2 it is the Bures-Wasserstein metric. We study two other cases: G(A, B) = A 1/2 (A −1/2 BA −1/2 ) 1/2 A 1/2 , the Pusz-Woronowicz geometric mean, and G(A, B) = exp log A+log B 2 , the log Euclidean mean. With these choices d(A, B) is no longer a metric, but it turns out that d 2 (A, B) is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of m positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy. √ 2 d(p, q) as the definition of the Hellinger distance. We have thenwhere A(p, q) is the arithmetic mean of the vectors p and q, G(p, q) is their geometric mean, and tr x stands for x i .A matrix/noncommutative/quantum version would seek to replace the probability vectors p and q by density matrices A and B ; i.e., positive semidefinite matrices A, B with tr A = tr B = 1. In the discussion that follows, the restriction on trace is not needed, and so we let A and B be any two positive semidefinite matrices. On the other hand, a part of our analysis 2010 Mathematics Subject Classification. 15B48, 49K35, 94A17, 81P45.

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On Wasserstein geometry of Gaussian measures

Cited by 115 publications

References 10 publications

Statistical Aspects of Wasserstein Distances

Statistical Aspects of Wasserstein Distances

Upscaling from particle models to entropic gradient flows

Matrix versions of the Hellinger distance

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