6.15 Image classication kernel. For  0, the kernel K : (x; x0) 7!

XN k=1 min(jxkj; jx0kj) (6.30)

over RN  RN is used in image classication. Show that K is PDS for all

 0. To do so, proceed as follows.

(a) Use the fact that (f; g) 7!

R +1 t=0 f(t)g(t)dt is an inner product over the set of measurable functions over [0;+1) to show that (x; x0) 7! min(x; x0) is a PDS kernel. (Hint: associate an indicator function to x and another one to x0.)

(b) Use the result from

(a) to rst show that K1 is PDS and similarly that K with other values of is also PDS.