# Statistical Consistency of Kernel Canonical Correlation Analysis

Kenji Fukumizu, Francis R. Bach, Arthur Gretton.

Year: 2007, Volume: 8, Issue: 14, Pages: 361−383

#### Abstract

While kernel canonical correlation analysis (CCA) has been applied
in many contexts, the convergence of finite sample estimates of the
associated functions to their population counterparts has not yet
been established. This paper gives a mathematical proof of the
statistical convergence of kernel CCA, providing a theoretical
justification for the method. The proof uses covariance operators
defined on reproducing kernel Hilbert spaces, and analyzes the
convergence of their empirical estimates of finite rank to their
population counterparts, which can have infinite rank. The result
also gives a sufficient condition for convergence on the
regularization coefficient involved in kernel CCA: this should
decrease as *n*^{-1/3}, where *n* is the number
of data.