The optimal rate of canonical correlation analysis for stochastic processes

Abstract Functional canonical correlation analysis (FCCA) has been applied in many contexts, but the asymptotic properties have not yet been studied enough. In this paper we consider a general setup resembling that of Eubank and Hsing (2008). We focus on the convergence of estimates of the associated canonical functions to their population counterparts. A regularized estimator is proposed based on the Tikhinov regularized approach. Under some conditions, an upper bound is derived for this estimator and furthermore, a sharp lower bound is established. Consequently, the minimax optimal rate is obtained and depends on the level of dependence between the two stochastic processes. A simple simulation is performed to illustrate our methods.