Canonical Correlation Analysis in high dimensions with structured regularization

Canonical Correlation Analysis (CCA) is a technique for measuring the association between two multivariate sets of variables. The Regularized modification of Canonical Correlation Analysis (RCCA) imposing $\ell_2$ penalty on the CCA coefficients is widely used in applications while conducting the analysis of high dimensional data. One limitation of such a regularization type is that it ignores the data structure treating all the features equally, which can be ill-suited for some applications. In this paper we introduce several approaches to regularizing CCA that take the underlying data structure into account. In particular, the proposed Group Regularized Canonical Correlation Analysis (GRCCA) is useful when the variables are grouped. We also suggest some tricks that allow to avoid excessive computations while conducting CCA with regularization in high dimensions. We demonstrate the applications of these methods to a small simulation example as well as to a real data example from neuroscience.