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Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

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Abstract

Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen–Loève 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. Biometrika101, 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 Fréchet 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.

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Acknowledgements

We wish to warmly thank a reviewer for providing constructive and insightful comments that led to genuine improvements in our presentation. This research is supported in part by a Swiss National Science Foundation grant to V. M. Panaretos.

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  1. Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Valentina Masarotto, Victor M. Panaretos & Yoav Zemel

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  1. Valentina Masarotto
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  2. Victor M. Panaretos
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  3. Yoav Zemel
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Correspondence to Victor M. Panaretos.

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Masarotto, V., Panaretos, V.M. & Zemel, Y. Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes. Sankhya A 81, 172–213 (2019). https://doi.org/10.1007/s13171-018-0130-1

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