Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

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 infinitedimensionality 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 Fréchet 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.

The Statistical Analysis of Acoustic Phonetic Data: Exploring Differences Between Spoken Romance Languages

“…Specifically, Masarotto et al . () show that the Procrustes distance coincides with the Wasserstein distance between centred Gaussian processes with corresponding covariances; and, consequently, that the X il can be seen to arise hierarchically, by first generating n latent realizations ɛ il ∼ IID N (0, C ) from the ‘Fréchet mean language’ with covariance C , and then deforming them by G group‐specific operators T l , i.e. X il = T l ɛ il .…”

“…I would like to remark that this choice of geometry implicitly imposes a concrete hierarchical generative model, with an appealing interpretation in the application's context, and potential implications on the analysis, either conceptual or concrete. Specifically, Masarotto et al (2018) show that the Procrustes distance coincides with the Wasserstein distance between centred Gaussian processes with corresponding covariances; and, consequently, that the X il can be seen to arise hierarchically, by first generating n latent realizations " il ∼ IID N.0, C/ from the 'Fréchet mean language' with covariance C, and then deforming them by G group-specific operators T l , i.e. X il = T l " il .…”

“…The maps {T l } can be recovered explicitly from knowledge of C and C l and essentially correspond to the log-images of the {C l } on the tangent space at the Fréchet mean C. In that sense, they can also be used to produce a tangent space principal component analysis of the main components of variation in the collection {C l }, potentially revealing further structure. See Masarotto et al (2018), sections 9 and 10.…”

Functional Models for Time-Varying Random Objects

“…Some of the examples given on the seventh page are already Hilbertian, and the same is true for the simulations and the data analysis. The authors also mention the Procrustes metric, which has an interpretation of a Wasserstein distance in finite or infinite dimensions (Masarotto et al, 2019). Another instance of a problem combining phase variation and discrete measurements on infinite dimensional and non-linear spaces is the registration of spatial point processes ).…”

Data analysis onnonstandardspaces

“…Masarotto et al (2019) observed that the Wasserstein L 2 ‐geometry is equivalent to a Procrustes‐like approach earlier proposed earlier by Dryden et al (2009), which is given by Here, the Cholesky factorization chol gives rise to a one‐to‐one map of SPD ( m ) to the full rank upper triangular matrices UT * ( m ) which embed in the upper triangular matrices UT ( m ), which map injectively into the space of centered configurations, which then project to Kendall's size‐and‐shape space.Remark (A Riemannian structure for the mildly rank deficient case) Upon close inspection, since the top dimensional stratum of comprises all configuration matrices a with rank( a ) ≥ m − 1 (that is where SO ( m ) acts freely), with this identification, the L 2 Wasserstein space of Gaussian measures which are at most degenerate in one dimension (or of positive semi‐definite matrices rank deficient in at most one dimension) have, via the injection ), the structure of the top‐dimensional incomplete Riemannian manifold stratum of , cf. Huckemann (2011a).…”