Barycenters in Alexandrov spaces of curvature bounded below

Ohta, Shin‐ichi

doi:10.1515/advgeom-2011-058

Cited by 29 publications

(29 citation statements)

References 23 publications

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“…The following inequality characterizes Alexandrov spaces with curvature bounded from below by zero [21]. Given a geodesic space X with metric d for any geodesic γ : [0, 1] → X from X to Y and any Z ∈ X…”

Section: Persistence Diagrams and Alexandrov Spaces With Curvature Bomentioning

confidence: 99%

Fréchet Means for Distributions of Persistence Diagrams

Turner

Mileyko

Mukherjee

et al. 2014

Discrete Comput Geom

210

266

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Abstract. Given a distribution ρ on persistence diagrams and observations X1, ...Xn iid ∼ ρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1, ...Xn. If the underlying measure ρ is a combination of Dirac masses ρ = 1 m m i=1 δZ i then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

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Section: Persistence Diagrams and Alexandrov Spaces With Curvature Bomentioning

confidence: 99%

Fréchet Means for Distributions of Persistence Diagrams

Turner

Mileyko

Mukherjee

et al. 2014

Discrete Comput Geom

210

266

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“…Precisely, any estimator of this class is associated with a suitable distance in the considered space and is defined as the geometric barycenter [13] of a set of covariance matrix, obtained from the available secondary data set. As to the considered distances, we focus on Euclidean, log-Euclidean [14], root-Euclidean, power-Euclidean and Cholesky distances.…”

Section: Introductionmentioning

confidence: 99%

Covariance matrix estimation via geometric barycenters and its application to radar training data selection

Aubry

Maio

Pallotta

et al. 2013

IET Radar, Sonar & Navigation

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This study deals with the problem of covariance matrix estimation for radar signal processing applications. The authors propose and analyse a class of estimators that do not require any knowledge about the probability distribution of the sample support and exploit the characteristics of the positive-definite matrix space. Any estimator of the class is associated with a suitable distance in the considered space and is defined as the geometric barycenter of some basic covariance matrix estimates obtained from the available secondary data set. Then, the authors introduce an adaptive detection structure, exploiting the new covariance matrix estimators, based on two stages. The former consists of a data selector screening among the training data, whereas the latter is a conventional adaptive matched filter taking the final decision about the target presence. At the analysis stage, the authors assess the performance of the proposed two-stage scheme in terms of probability of correct outliers excision, constant false alarm rate behaviour and detection probability. The analysis is conducted both on simulated data and on the challenging KASSPER datacube.

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“…In spaces with Alexandrov curvature bounded above, the behaviour of barycenters is fairly well understood; see the work of Sturm [28]. The study of barycenters on spaces with curvature bounded below has recently been initiated by Ohta, and remains in its infancy [21]. It is, however, already apparent that barycenters on spaces with lower curvature bounds are not as well behaved as their counterparts on spaces with upper curvature bounds.…”

Section: Introductionmentioning

confidence: 99%

Optimal transportation with infinitely many marginals

Pass

2013

Journal of Functional Analysis

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We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Swiech [14]. We prove results on the existence, uniqueness and characterization of the optimizer, which are natural extensions of the results in [14]. The proof relies on a relationship between this problem and the problem of finding barycenters in the Wasserstein space, a connection first observed for finitely many marginals by Agueh and Carlier [1]. M1×M2c(x 1 , x 2 )dγ Equivalently, one can formulate this problem using more probabilistic language. Here one looks for an M 1 × M 2 valued random variable (X 1 , X 2 ), such that lawX i = µ i , for i = 1, 2, which minimizes the expectation:

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Barycenters in Alexandrov spaces of curvature bounded below

Cited by 29 publications

References 23 publications

Fréchet Means for Distributions of Persistence Diagrams

Fréchet Means for Distributions of Persistence Diagrams

Covariance matrix estimation via geometric barycenters and its application to radar training data selection

Optimal transportation with infinitely many marginals

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