11.4 Perturbed kernels. Suppose two dierent kernel matrices, K and K0, are used to train two kernel ridge regression hypothesis with the same regularization parameter . In this problem, we will show that the dierence in the optimal dual variables, and 0 respectively, is bounded by a quantity that depends on kK0 ????Kk2.

(a) Show 0 ???? =

????

(K0 + I)????1(K0 ???? K)(K + I)????1

y. (Hint: Show that for any invertible matrix M, M01 ????M1 = ????M0????1(M0 ????M)M????1.)

(b) Assuming 8y 2 Y; jyj  M, show that k0 ???? k 

pmMkK0 ????Kk2

2 :