machine learning - Kernel Addition and one Surprisingly facts? -

if k1 , k2 kernel in space r^n*r^n know k(x,z)=ak1(x,z) + bk2(x,z) (kernel addition) still kernel (valid kernel) if a,b >= 0 (a,b real numbers, scalar) . valid can seen fact results of kernel function can interpreted inner products in feature space. adding inner products amounts adding both feature spaces together. today i'm surprised when read notes. ta says k(x,z)=ak1(x,z) + bk2(x,z) (kernel addition) still kernel if a<0,b>0 or a>0,b<0 !!!

is possible still there valid kernel a<0,b>0 , a>0,b<0? expert me ?

a kernel should non-negative equal inputs. let's denote arbitrary kernel (rbf, example) k.

let's consider k1 = k2 = k.

define k3 = k1 + b k2. let a = -2, b = 1. k3(x, x) = -k(x, x), fails satisfy kernel's definition.

now, example above show statement of ta not true in general, k1, k2, a , b. there are, obviously, combinations of these result in proper kernel.

i suppose, necessary condition λmin(a k1) + λmin(b k2) >= 0 λmin means smallest eigenvalue of operator. λmin(c k) equal c λmin(k) non-negative c, , c λmax(k) negative one. consider a=2, b=-1 same kernels above example.

in case of both a < 0 , b < 0, though, no valid kernel can forged. easy see: if k3(x, x) > 0, then, k3(x, x) = k1(x, x) + b k2(x, x) = -|a| k1(x, x) - |b| k2(x, x) > 0 since a , b negative. leads |a| k1(x, x) + |b| k2(x, x) < 0 can't true since k1 , k2 valid kernels.