Archive / INF Seminars / INF_2024_03_14_Zemel_Yoav
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Optimal transport for covariance operators


Host: Prof. Ernst-Jan Camiel Wit




USI Campus Est, room D1.14, Sector D
13:45 - 14:30

Yoav Zemel
EPFL, Switzerland
We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure, and describing the main modes of variation in this second order structure, if present. These tasks can be seen as an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators, respectively. They arise naturally in functional data analysis, where several populations are to be contrasted relative to the nature of their dispersion around their means, rather than relative to their means themselves. We contribute a novel approach based on optimal (multi)transport, where each covariance can be identified with a a centred Gaussian process of corresponding covariance. By means of constructing the optimal simultaneous coupling of these Gaussian processes, we contrast the (linear) maps that achieve it with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, is seen to distinctly outperform the state-of-the-art, and to furnish considerable power even under local alternatives. This effect is seen to be genuinely functional, and is related to the potential for perfect discrimination in infinite dimensions. In the event of a rejection of the null hypothesis stipulating equality, a geometric interpretation of the transport maps allows us to construct a (tangent space) PCA revealing the main modes of variation. As a necessary step to developing our methodology, we prove results on the existence and boundedness of optimal multitransport maps. These are of independent interest in the theory of transport of Gaussian processes.

Yoav Zemel is Bernoulli Instructor at École polytechnique fédérale de Lausanne (EPFL), working on applications of optimal transport to various statistical problems. He obtained his PhD at EPFL in 2017. Afterwards, he was an SNSF postdoc at the University of Göttingen and then a postdoc at the University of Cambridge, before returning to EPFL as Bernoulli Instructor in 2022.

Host: Prof. Ernst-Jan Camiel Wit