Regularized Functional Canonical Correlation Analysis for Stochastic Processes

TL;DR

This paper derives the asymptotic distributions of regularized estimators for functional canonical correlation, focusing on RKHS-based methods and analyzing Tikhonov and TSVD regularizations for stochastic processes.

Contribution

It provides the first theoretical analysis of the asymptotic behavior of regularized functional canonical correlation estimators in RKHS frameworks.

Findings

Asymptotic distributions derived for regularized estimators

Comparison of Tikhonov and TSVD regularization methods

Theoretical justification for RKHS-based functional CCA

Abstract

In this paper we derive the asymptotic distributions of two distinct regularized estimators for functional canonical correlation as well as their associated eigenvalues, eigenvectors and projection operators. The methods we developed utilize regularized estimators which approach the functional operators based in reproducing kernel Hilbert spaces (RKHS) as the regularization parameter approaches zero. In addition to providing some justification for the RKHS methods, we explore the asymptotics of regularized operators associated with both Tikhinov and truncated singular value decomposition (TSVD) type regularization. Together, these regularization methods represent two of the most commonly utilized forms of regularization.