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Statistical consistency of kernel canonical correlation analysis. (English) Zbl 1222.62063
Summary: 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.

##### MSC:
 62H20 Measures of association (correlation, canonical correlation, etc.) 46N30 Applications of functional analysis in probability theory and statistics
##### Keywords:
regularization; Hilbert space
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