6.18 Metrics and Kernels. Let X be a non-empty set and K: XX ! R be a negative denite symmetric kernel such that K(x; x) = 0 for all x 2 X.

(a) Show that there exists a Hilbert space H and a mapping (x) from X to H such that:

K(x; y) = jj(x) ???? (x0)jj2 :

Assume that K(x; x0) = 0 ) x = x0. Use theorem 6.16 to show that pK denes a metric on X.

(b) Use this result to prove that the kernel K(x; y) = exp(????jx????x0jp), x; x0 2 R, is not positive denite for p > 2.

(c) The kernel K(x; x0) = tanh(a(x
x0) +

b) was shown to be equivalent to a two-layer neural network when combined with SVMs. Show that K is not positive denite if a < 0 or b < 0. What can you conclude about the corresponding neural network when a < 0 or b < 0?