We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure. This task can be seen as an Analysis of Variance (ANOVA) of covariance operators, and arises naturally in functional data analysis. We formulate the problem based on the nonlinear geometry of multimarginal transport, where each covariance can be identified with a a centred Gaussian process. By suitably contrasting the optimal multimarginal transport operators to the identity, it is seen that one can distinctly outperform existing tests, with considerable power even under local alternatives. This effect is seen to be genuinely functional, and we conclude by showing how can can harness this functional phenomenon in order to construct powerful tests in more traditional settings. Based on joint work with Valentina Masarotto (Leiden), Leonardo Santoro (EPFL), Yoav Zemel (EPFL), and Kartik Waghmare (ETH Zürich).
