Title: Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

Author(s): Valentina Masarotto, Victor M. Panaretos and Yoav Zemel
Issue: Volume 81 Series A Part 1 Year 2019
Pages: 172 -- 213
Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen–Lo`eve 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. In this paper, 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 (Pigoli et al. Biometrika 101, 409–422, 2014). 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 detailed description of those aspects of this manifold-like geometry that are important in terms of statistical inference; to establish key properties of the Fr´echet mean of a random sample of covariances; and to define generative models that are canonical for such metrics and link with the problem of registration of warped functional data.
Primary 60G15 Gaussian processes, 60D05 Geometric probability and stochastic geometry; Secondary 60H25 Random operators and equations, 62M99 None of the above, but in this section.
Keywords and phrases: Functional data analysis, Fr´echet mean, Manifold statistics, Optimal coupling, Tangent space PCA, Trace-class operator.