Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

TL;DR

This paper explores the geometric structure of covariance operators in infinite-dimensional spaces using Procrustes and Wasserstein metrics, linking them to Gaussian processes and optimal transportation for advanced statistical analysis.

Contribution

It introduces a geometric framework for covariance operators using Procrustes and Wasserstein metrics, connecting them to Gaussian processes and functional data registration.

Findings

Characterizes the manifold geometry of covariance operators.

Establishes properties of the Fréchet mean in this space.

Links covariance operator analysis with optimal transportation theory.

Abstract

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 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 geometry that are important in terms of statistical inference, and establish key properties of the Fr\'echet 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.