Canonical correlation analysis for infinite dimensional data

This work deals with the analysis of the notion of Canonical Correlation Analysis for infinite dimensional data.The concept of a squared correlation is extended to an infinite dimensional setting.An interesting observation is that the definition for the finite dimensional case, that is, the Euclidean case, breaks down for the infinite dimensional setting.Therefore, we introduce a regularization technique to deal with the problem.We show that regularization is indispensable for the infinite dimensional case.We provide a consistency result for the population squared principal canonical correlation. Asymptotics for the sample squared principal canonical correlation and corresponding canonical variates are established through a delta-method for analytic functions of covariance operators aided by some perturbation theory.