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*
λ
in (10) is quadratic programming (QP) problem.
There are many approaches to sole this problem; for instance, the sequential
minimal optimization is described as below:
However the way to find
A QP with two variables is trivial to solve
λ
,
λ
At each iteration, a pair of (
) is picked up and QP is solved with these
i
j
variables; repeat until convergence
2.2 Applying Support Vector Machine into Document
Classification
Give corpus D = { D 1 , D 2 , D 3 ,…, D m } in which every document D i is modeled by a
tf-idf weight vector. Suppose there are n terms { t 1 , t 2 ,…, t n }, we have:
D i = ( d i1 , d i2 ,…, d in )
where d ij is product of term frequency and inverse document frequency d ij =
tf ij * idf j .
If the index of document is ignored, document D is represented as below:
D = ( d 1 , d 2 ,…, d n )
Given k classes { l 1 , l 2 ,…, l k }, there is demand of classifying documents into such
classes. The technique of classification based on SVM is two-class classification
in which the classes is +1, -1 for y i =+1, -1 respectively. So we need to determine
unique hyper-plane referred to as two-class classifier. It is possible to extend two-
class classification to k-class classification by constructing k two-class classifier.
In means that we must specify k couple of optimal weight vector W i * and bias b i * .
Each couple ( W i * , b i * ) being a two-class classifier is the representation of class l i .
The process of finding ( W i * , b i * ) in training corpus D is described in the section of
support vector machine.
Table 1 k couple ( W i * , b i * ) corresponds with k class { l 1 , l 2 ,…, l k }
Class
Weight vector
Bias
Classification rule
W 1 *
b 1 *
R 1 = W 1 *T
l 1
X + b 1 *
W 2 *
b 2 *
l 2
R 2 = W 2 *T
X + b 2 *
W k *
b k *
R k = W k *T
l k
X + b k *
For example, classifying document D= ( d 1 , d 2 ,…, d n ) is described as below:
1. For each classification rule R i = W i *T
X + b i * , substituting each D into such
rule. It means that vector X in such rule is replaced by document D .
Expression i = W i *T
D + b i *
2. Suppose there is a sub-set of rules { R h+1 , R h+2 ,…, R h+r } that the value of
expression Expression i = W i *T
D + b i * is greater than or equal 1 . We can
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