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K ( x i , x j )= x i · x j
· x j ) d
K ( x i , x j )=(1+ x i
K ( x i , x j )= e −γ x i − x j 2
K ( x i , x j )=tanh( θ 1 x i
· x j −
θ 2 )
Now we can define the decision rule used by the system. The classification
algorithm has to predict the value P ( w ) of the predicate P for a given word w .
The corresponding decision rule is
Decide P ( w )= 1 , if g ( f ( w ))
0 , otherwise .
Evaluation of the System
To test and evaluate our pattern recognition system we generate several test
datasets of different types:
A test set S e which is generated by the same procedure as for the training
set D , but independently of D .
A test set S R of pseudo-randomly generated cyclically reduced elements of
F ( X ),asdescribedinSection3.1.
A test set S P of pseudo-randomly generated cyclically reduced primitive
elements in F ( X ). Recall that w
F ( X ) is primitive if and only if there
exists a sequence of Whitehead automorphisms t 1 ...t m ∈
Ω ( X ) such that
X ± 1 .Elementsin S P are generated by the
procedure described in , which, roughly speaking, amounts to a random
choice of x
t m ...t 1 ( x )= w for some x
X ± 1
and a random choice of a sequence of automorphisms
t 1 ...t m ∈
Ω ( X ).
A test set S 10 which is generated in a way similar to the procedure used
to generate the training set D . The only difference is that the non-minimal
elements are obtained by applying not one, but several randomly chosen
automorphisms from Ω ( X ). The number of such automorphisms is chosen
uniformly randomly from the set
1 ,..., 10
, hence the name.
For more details on the generating procedure see .
To show that performance of Support Vector Machines is acceptable for free
groups, including groups of large ranks, we run experiments with groups of ranks
3,5,10,15,20. For each group we construct the training set D and test sets S e , S 10 ,
S R , S P using procedures described previously. Some statistics of the datasets are
given in Table 1.