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In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers as equivalent a

Procrustes Metrics and Optimal Transport for Covariance Operators

Valentina Masarotto

Covariance operators play a fundamental role 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 the space of such operators, as well as with their intrinsic infinite-dimensionality. We will show that we can identify the space of infinite-dimensional covariance operators equipped with the Procrustes size-and-shape metric from shape theory, with that of centred Gaussian processes, equipped with the Wasserstein metric of optimal transportation. We then describe key geometrical and topological aspects of the space of covariance operators endowed with the Procrustes metric. Through the notion of multicoupling of Gaussian measures, we establish existence, uniqueness and stability for the Fréchet mean of covariance operators with respect to the Procrustes metric. Furthermore, we will provide generative models that are canonical for such metric. We then turn to the problem of comparing several samples of stochastic processes with respect to their second-order structure, and we subsequently describe the main modes of variation in this second order structure. These two tasks are carried out via an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators respectively. In order to perform ANOVA, we introduce a novel approach based on optimal (multi)transport and identify each covariance with an optimal transport map. These maps are then contrasted with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, outperforms the state-of-the-art in the functional case. If the null hypothesis postulating equality of the operators is rejected, thanks to a geometric interpretation of the transport maps we can construct a PCA on the tangent space with the aim of understanding the sample variability. Finally, we provide a further example of use of the optimal transport framework, by applying it to the problem of clustering of operators. Two different clustering algorithms are presented, one of which is innovative. The transportation ANOVA, PCA and clustering are validated both on simulated scenarios and real dataset.

EPFL2019

Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

Valentina Masarotto, Victor Panaretos, Yoav Zemel

Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loeve 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 Frechet 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.

2019

Procrustes Metrics and Optimal Transport for Covariance Operators

Valentina Masarotto

EPFL2019