6.11 Explicit mappings.

(a) Denote a data set x1; : : : ; xm and a kernel K(xi; xj) with a Gram matrix K. Assuming K is positive semidenite, then give a map (
) such that K(xi; xj) = h(xi); (xj)i.

(b) Show the converse of the previous statement, i.e., if there exists a mapping

(x) from input space to some Hilbert space, then the corresponding matrix K is positive semidenite.