Abstract: We develop a framework of canonical correlation analysis for distribution-valued functional data within the geometry of Wasserstein spaces. Specifically, we formulate an intrinsic concept of correlation between random distributions, propose estimation methods based on functional principal component analysis and Tikhonov regularization, respectively, for the correlation and its corresponding weight functions, and establish the minimax convergence rates of the estimators. In order to overcome the challenge raised by nonlinearity ofWasserstein spaces, the key idea is to adopt tensor Hilbert spaces to distribution-valued functional data. The finite-sample performance of the proposed estimators is illustrated via simulation studies, and the practical merit is demonstrated via a study on the association of distributions of brain activities between brain regions.

Key words and phrases: Distribution-valued data, minimax rate, parallel transport, tensor Hilbert space.