Worksheet (non-assessed)
Support vector machines
a) What is the main idea behind linear Support Vector Machines (SVM)? Illustrate your
explanation by drawing a figure.
b) How can the linear SVM be made non-linear?
c) The following kernels are examples of proper kernels:
K1 ( x1 , x2 )  e

|| x1  x 2 || 2
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K 2 ( x1 , x2 )  x1  x2  2
We know that:
- given a proper kernel K, then aK, a > 0, is a proper kernel
- given proper kernels K' and K'', then K' K'' is a proper kernel
- if K is a proper kernel, for any real valued function f (x) over the input space,
K ( x1, x2 )  f ( x1 ) f ( x2 ) is a proper kernel.
Decide which of the following formulae define proper kernels and explain why.
K 3 ( x1 , x2 )  4 K 2 ( x1 , x2 )  K1 ( x1 , x2 )
K 4 ( x1 , x2 )  K1 ( x1 , x2 ) 2 K 2 ( x1 , x2 )
K 5 ( x1 , x2 )  5 K1 ( x1 , x2 )
d) Consider the 2-dimensional inputs xi  ( x1i , x2i )   2 .
Is the following a proper kernel? Explain why.
K ( x1, x2 )  (( x11 )2  5x12 )(( x12 )2  5x22 )