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A0298
Title: Procrustes metrics on covariance operators and optimal transport of Gaussian processes Authors:  Yoav Zemel - Georg-August Universität Göttingen (Germany) [presenting]
Victor Panaretos - EPFL (Switzerland)
Valentina Masarotto - EPFL (Switzerland)
Abstract: Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loeve 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 infinite-dimensionality of these operators. We describe the manifold-like 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-like geometry that are important in terms of statistical inference, and establish key properties of the Frechet 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.