Support Vector Machines
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- tags
- Machine Learning
Support vector machines were (and still are) a popular technique for determining a linear decision boundary between two classes. This can be extended to more classes and regression through Reference kernel machines, but this note only goes through the basics of the two class approach. We will also assume that the data is already linearly separable, and will discuss if there is overlapping classes in the separable space in svm:overlapping and some useful data transformations for SVMs in
Lets start with the problem specification. We are given a data set of \(N\) input vectors \(\xvec_1, \ldots, \xvec_N\) with corresponding target values \(t_1, \ldots, t_N\), where the target only takes values in \(t_n \in \{-1, 1\}\). This is 2-class Classification. We will first work in feature-space (using a fixed transformation \(\phi\)) and then move to functional-space with kernels later. We want to build a discriminatory
\[y(\xvec) = \wvec^\trans \phi(\xvec) + b\]
such that \(t_n y(\xvec_n) > 0\) for all training data points (i.e. the function \(y\) discriminates between classes by its sign not by its magnitude).
The solution (also known as the decision boundary) chosen by the SVM is the one for which the margin is maximized.
The margin is defined as the smallest distance between the decision boundary and any of the samples. The Perpendicular Distance between the point \(\xvec\) and the hyper-plane defined by \(y(\xvec)=0\) where \(y(\xvec)\) is of the linear form above is \(\frac{|y(\xvec)|}{\lVert \wvec \rVert}\).