Machine Learning : Support Vector Machines
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1 Machine Learning Support Vector Machines 05/01/2014 Machine Learning : Support Vector Machines
2 Linear Classifiers (recap) A building block for almost all a mapping, a partitioning of the input space into half-spaces that correspond to classes. Decision rule: is the normal to the hyper plane (Synonyms Neuron model, Perceptron etc.) 05/01/2014 Machine Learning : Support Vector Machines 2
3 Two learning tasks Let a training dataset be given with (i) data and (ii) classes The goal is to find a hyper plane that separates the data (correctly) Now: The goal is to find a corridor (stripe) of the maximal width that separates the data (correctly). 05/01/2014 Machine Learning : Support Vector Machines 3
4 Linear SVM Remember that the solution is defined only up to a common scale Use canonical (with respect to the learning data) form in order to avoid ambiguity: The margin: The optimization problem: 05/01/2014 Machine Learning : Support Vector Machines 4
5 Linear SVM The Lagrangian of the problem: The meaning of the dual variables v : a) (a constraint is broken) maximization wrt. gives: (surely not a minimum) b) maximization wrt. gives no influence on the Lagrangian c) does not mater, the vector is located on the wall of the corridor Support Vector 05/01/2014 Machine Learning : Support Vector Machines 5
6 Linear SVM Lagrangian: Derivatives: The solution is a linear combination of the data points. 05/01/2014 Machine Learning : Support Vector Machines 6
7 Linear SVM Substitute into the decision rule and obtain the vector is not needed explicitly!!! The decision rule can be expressed as a linear combination of scalar products with support vectors. Only strictly positive (i.e. those corresponding to the support vectors) are necessary for that. 05/01/2014 Machine Learning : Support Vector Machines 7
8 Linear SVM Substitute into the Lagrangian and obtain the dual task can also be expressed in terms of scalar products only, the data points are not explicitly necessary. 05/01/2014 Machine Learning : Support Vector Machines 8
9 Feature spaces 1. The input space is mapped onto a feature space by a nonlinear transformation 2. The data are separated (classified) by a linear decision rule in the feature space Example: quadratic classifier The transformation is (the images are separable in the feature space) 05/01/2014 Machine Learning : Support Vector Machines 9
10 Feature spaces The images are not explicitly necessary in order to find the separating plane in the feature space, but their scalar products For the example above: the scalar product can be computed in the input space, it is not necessary to map the data points onto the feature space explicitly. Such functions are called Kernels. 05/01/2014 Machine Learning : Support Vector Machines 10
11 Kernels Kernel is a function feature space that computes scalar product in a Neither the corresponding space nor the mapping need to be specified thereby explicitly Black Box. Alternative definition: if a function is a kernel, then there exists such a mapping, that The corresponding feature space is called the Hilbert space induced by the kernel. Let a function Mercer s theorem. be given. Is it a kernel? 05/01/2014 Machine Learning : Support Vector Machines 11
12 Kernels Let and be two kernels. Than are kernels as well (there are also other possibilities to build kernels from kernels). Popular Kernels: Polynomial: Sigmoid: Gaussian: (interesting : ) 05/01/2014 Machine Learning : Support Vector Machines 12
13 An example The decision rule with a Gaussian kernel 05/01/2014 Machine Learning : Support Vector Machines 13
14 Conclusion SVM is a representative of discriminative learning i.e. with all corresponding advantages (power) and drawbacks (overfitting) remember e.g. the Gaussian kernel with The building block linear classifiers. All formalisms can be expressed in terms of scalar products the data are not needed explicitly. Feature spaces make non-linear decision rules in the input spaces possible. Kernels scalar product in feature spaces, the latter need not be necessarily defined explicitly. Literature (names): Bernhard Schölkopf, Alex Smola... Nello Cristianini, John Shawe-Taylor... 05/01/2014 Machine Learning : Support Vector Machines 14
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