Illustration of CCA. The left column depicts point clouds of two-dimensional views $X$ and $Y$ with lines indicating the span of the top principal component and top canonical correlation basis vector for each view (labeled “PCA” and “CCA,” respectively). The center and right columns depict point clouds of joint one-dimensional projections of $X$ and $Y$ onto their top principal component or top canonical correlation basis vector, with the correlations between the two projections listed. As illustrated in the lower right plot, the correlation between the projected views is maximized when each view is projected onto its top canonical correlation basis vector.