A Kernelbased Support Vector Machine

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A Kernel-based Support Vector Machine
byPeter Axelberg and Johan Löfhede
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Maximum Margin Classifier
?1
Maximum Margin
Decision boundary
?2
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Need for more complex decision boundaries

• In general, real-world applications require
  more complex decision boundaries than linear
  functions.

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Dimension expansion

• The SVM offers a method where the input space
  is mapped by a non-linear function, , to a
  higher dimensional feature space where the
  classes are more likely to be linearly separable.

?(x)
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High dimensional feature space
?(Xi)
f(?(Xi))
?(Xi)
xi
?(x1)
?1
x1
?2
Input space
Decision space
High dimensional feature space
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Kernel function
?(x)
Instead of directly using the mapping vectors
in the high dimentional feature space, a
kernel function K(x,xi) can be introduced in the
input space according to the following
substitution
where lt gt denotes the inner product
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The Kernel function used by the decision boundary
function
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Examples of Kernel functions

• Polynomial
• Radial basis function (RBF)
• Sigmoidal

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A real world example

• Classification of some fruits/vegetables using
  kernel-based SVM

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