Optimal Transport for Covariance Operators

Santoro Leonardo, EPFL

Covariance operators are increasingly becoming the main target of statistical interest in sev- eral contexts, but the infinite-dimensional nature and intrinsic non-linear structure of covariances renders their statistical analysis challenging. An important model for the peculiar geometric structure of covariances is the Bures-Wasserstein metric, which identifies this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation, with connections to quantum information theory and shape analysis. We establish key prop- erties of the Fr ́echet mean of a random sample of covariances, such as existence, uniqueness, regularity and large sample theory, and make use of the manifold-like structure of this space to develop a PCA, relating to the stability of the optimal transportation problem. Our develop- ment is genuinely infinite dimensional, highlighting how properties that behave predictably in finite dimensions turn unexpectedly in the general infinite-dimensional case.

Area: CS47 - Statistical inference in infinite-dimensional spaces (Alessia Caponera)

Keywords: covariance operator, Frechet mean, optimal transport, bures-wasserstein

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