Regularization

A second method considers regularizing the solution by adding a normalization restriction to the norm of $\textbf{\~h}_k$ [11,12] or $\alpha$$_k$ [21]. Here, we apply the classical regularization technique in the framework of LS regression, which yields the following CCA-GEV problem

$$\frac{1}{2} \begin{bmatrix}\textbf{X}_1^T \textbf{X}_1 & \textbf{... ...tbf{0} & \textbf{X}_2^T \textbf{X}_2 + c \textbf{I} \end{bmatrix} \textbf{\~h},$$

or, in the case of K-CCA

$$\frac{1}{2} \begin{bmatrix}\textbf{K}_1 & \textbf{K}_2\\ \textbf{K}_1 & \textbf{K}_2 \end{bmatrix} \mbox{\boldmath$\alpha$\unboldmath } = \beta \begin{bmatrix}\textbf{K}_1 + c {\mathbf I}& \textbf{0}\\ \textbf{0} & \textbf{K}_2 + c {\mathbf I} \end{bmatrix} \mbox{\boldmath$\alpha$\unboldmath } .$$ (7)

Therefore, the regularized version of CCA (or K-CCA) can be reformulated as two coupled LS (or kernel-LS) regression problems.