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统计学习精要 (Elements of Statistical Learning ) 习题 3.20

问题：Consider the canonical-correlation problem (3.67). Show that the leading pair of canonical variates











u






1















$u_1$
and











v






1















$v_1$
solve the problem

$$\underset{u^T(Y^T Y )u = 1\\ v^T (X^T X)v =1 }\max u^T(Y^T X )v,$$

a generalized SVD problem. Show that the solution is given by











u






1







=


(





Y








T









Y







)








−





1






2























u






∗






1















$u_1 = (Y^T Y)^{-\frac{1}{2}} u_1^*$
, and











v






1







=


(





X








T









X







)








−





1






2























v






∗






1















$v_1 = (X^T X)^{-\frac{1}{2}} v_1^*$
, where











u






∗






1















$u_1^*$
and











v






∗






1















$v_1^*$
are the leading left and right singluar vectors in