Analysis of Text Patterns Using Kernel Methods - Text Mining: Classification, Clustering, and Applications - page 9

The finitely positive semi-definite property completely characterizes ker-
nels because it is possible to construct the feature space assuming only this
property. The result is stated in the form of a theorem.

THEOREM 1.1 Characterization of kernels
Afunction

κ : X × X −→ R

can be decomposed

intoafeaturemapφ into a Hilbert space F applied to both its arguments
followed by the evaluation of the inner product in F if and only if it satisfies
the finitely positive semi-definite property.

κ ( x , z )= φ ( x ) ,φ ( z )

A preliminary concept useful to outline the Mercer's Theorem is the follow-
ing.
Let L 2 ( X ) be the vector space of square integrable functions on a compact
subset X of

R

n

with the definitions of addition and scalar multiplication;

formally

L 2 ( X )= f :
X

.

f ( x ) 2 dx < ∞

For mathematical details see ( 27 ).

THEOREM 1.2 Mercer
Let X be a compact subset of

n .Supposeκ is a continuous symmetric
function such that the integral operator T κ : L 2 ( X )

R

→ L 2 ( X ) ,

( T κ f )( · )=

κ ( ·, x ) f ( x ) d x

X

is positive, that is

κ ( x , z ) f ( x ) f ( z ) d x d z ≥ 0

X×X

for all f L 2 ( X ) . Then we can expand κ ( x , z ) in a uniformly convergent

series (on X

∈

× X) in terms of functions φ j , satisfying φ i ,φ j = δ ij :

κ ( x , z )= ∞

φ j ( x ) φ j ( z ) .

j =1

Furthermore, the series i =1

2

φ i

L 2 ( X ) is convergent.

The conditions of Mercer's Theorem are equivalent to requiring that for
every finite subset of X , the corresponding matrix is positive semi-definite
(6).