A Kernel-based Support Vector Machine by Peter Axelberg and Johan Löfhede

Maximum Margin Classifier Decision boundary 22 11 Maximum Margin

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

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)(x)

Input space High dimensional feature space x1x1 (x1)(x1) (Xi)(Xi) Decision space 11 22 xixi (Xi)(Xi) f(  ( X i )) High dimensional feature space

Kernel function Instead of directly using the mapping vectors in the high dimentional feature space, a kernel function K(x,x i ) can be introduced in the input space according to the following substitution: (x)(x) where denotes the inner product

The Kernel function used by the decision boundary function

Examples of Kernel functions  Polynomial  Radial basis function (RBF)  Sigmoidal

A real world example  Classification of some fruits/vegetables using kernel-based SVM

Feature 1Feature 2Feature 3Feature 4

Circumference 1 (longest) (cm) Circumference 2 (shortest) (cm) Weight (g)Color (code)