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where
φ
is a mapping from
X
to an (inner product) feature space
F
φ
:
x
−→
φ
(
x
)
∈
F.
x
and
z
can be elements of any set, and in this chapter they will be text
documents. Clearly, the image
φ
(
x
) is a vector in
N
.
R
n×n
Kernel Matrix.
The square matrix
K
∈
R
such that
K
ij
=
κ
(
x
i
,
x
j
)
for a set of vectors
{
x
1
,...,
x
n
}⊆
X
and some kernel function
κ
is called
kernel matrix
.
Modularity.
As we pointed out, the kernel component is data specific, while
the pattern analysis algorithm is general purpose. Similarly, substituting a
different algorithm while retaining the chosen kernel leads us to perform a
different type of pattern analysis. Clearly, the same kernel function or algo-
rithm can be suitably reused and adapted to very different kinds of problems.
Figure 1.1 shows the stages involved in the implementation of a typical kernel
approach analysis. The data are processed using a kernel to create a kernel
matrix, which in turn is processed by a pattern analysis algorithm to obtain a
pattern function. This function will be used to understand unseen examples.
FIGURE 1.1
: Modularity of kernel-based algorithms: the data are trans-
formed into a kernel matrix, by using a kernel function; then the pattern
analysis algorithm uses this information to find interesting relations, which
are all written in the form of a linear combination of kernel functions.
Using ecient kernels, we can look for linear relations in very high dimen-
sional spaces at a very low computational cost. If it is necessary to consider
a non-linear map
φ
, we are still provided with an ecient way to discover
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