Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

Kernel methods 147

DEFINITION 4.1 A valid kernel is a function κ that, for all g, g

′

∈ IR

N

,

satisfies

κ(g, g

′

) = φ(g)

⊤

φ(g

′

), (4.23)

where φ is given by Equation (4.21). For a set of observations g(ν), ν =

1 . . . m, the m × m matrix K with elements κ(g(ν), g(ν

′

)) is called the kernel

matrix.

Note that with Equation (4.22) we can w rite the kernel matrix equivalently

in the form

K = ΦΦ

⊤

. (4.24)

In Section 2.6.4 we saw that the Gram matrix is symmetr ic and posi-

tive semi-definite. This is also the case for kernel matrices since, fo r any

m-component vector x,

x

⊤

Kx =

X

ν,ν

′

x

ν

x

ν

′

K

νν

′

=

X

ν,ν

′

x

ν

x

ν

′

φ

⊤

ν

φ

ν

′

= (

X

ν

x

ν

φ

ν

)

⊤

(

X

ν

′

x

ν

′

φ

ν

′

)

=





X

ν

x

ν

φ

ν





2

≥ 0.

Positive semi-definiteness of the kernel matrix is in fact a necessary and suffi-

cient condition for any symmetric function κ