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Kernel Matrix as an Interface. All the information that is required by
the pattern analysis algorithm is inside the kernel matrix. The kernel matrix
can be seen as an interface between the input data and the pattern analysis
algorithm (see Figure 1.1 ), in the sense that all the data information passes
through the bottleneck. Several model adaptations and selection methods are
implemented by manipulating the kernel matrix. This property in some sense
is also a limitation, because if the kernel is too general no useful relation can
be highlighted in data.
1.2.3 Operations on Kernel Functions
As we pointed out, the positive semi-definiteness property is the core for
the characterization of kernel functions. New functions are kernels if they are
finitely positive semi-definite. So it is sucient to verify that the function
is a kernel and this demonstrates that there exists a feature space map for
which the function computes the corresponding inner product. It is important
to introduce some operations on kernel functions which always give as result
a new positive semi-definite function. We will say that the class of kernel
functions is closed under such operations.
The following two propositions can be viewed as showing that kernels satisfy
a number of closure properties, allowing us to create more complicated kernels
from simpler ones.
PROPOSITION 1.1 Closure properties
Let κ 1 and κ 2 be kernels over X
×
X, X
R
n , a
R
+ , f (
·
) a real-valued
function on X, φ : X
N ,and B a
symmetric positive semi-definite n × n matrix. Then the following functions
are kernels:
−→ R
N
with κ 3 a kernel over
R
N
× R
1. κ ( x , z )= κ 1 ( x , z )+ κ 2 ( x , z )
2. κ ( x , z )= 1 ( x , z )
3. κ ( x , z )= κ 1 ( x , z ) κ 2 ( x , z )
4. κ ( x , z )= f ( x ) f ( z )
5. κ ( x , z )= κ 3 ( φ ( x ) ( z ))
6. κ ( x , z )= x Bz with x , z
X
PROPOSITION 1.2
Let κ 1 ( x , z ) be a kernel over X × X,where x , z ∈ X,andp ( x ) a polynomial
with positive coe cients. Then the following functions are also kernels:
1. Polynomial kernel (4):
κ ( x , z )= p ( κ 1 ( x , z ))
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