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LINEAR CLASSIFICATION Motivation Suppose we want to classify some data points into two classes. Often we are interested in classifying data as part of a machine-learning process. These data points may not necessarily be points in R2 but may be multidimensional points. We are interested in whether we can separate them by a Hyperplane (a generalization of a Plane in three dimensional space to more than three dimensions). As we examine a hyperplane, this form of classification is known as linear classification. We also want to choose a hyperplane that separates the data points "neatly", with maximum distance to the closest data point from both classes -- this distance is called the ''margin''. We desire this property since if we add another data point to the points we already have, we can more accurately classify the new point since the separation between the two classes is greater. Now, if such a hyperplane exists, the hyperplane is clearly of interest and is known as the '' Maximum-margin Hyperplane '' or the ''optimal hyperplane'', as are the vectors that are closest to this hyperplane, which are called the ''support vectors''. Formalization We consider data points of the form: where the ''c''''i'' is either 1 or −1 -- this constant denotes the class to which the point ''x''''i'' belongs. We can view this as ''training data'', which denotes the correct classification which we would like the SVM to eventually distinguish, by means of the dividing hyperplane, which takes the form : As we are interested in the maximum margin, we are interested in the support vectors and the parallel hyperplanes (to the optimal hyperplane) closest to these support vectors in either class. It can be shown that these parallel hyperplanes can be described by equations : : | ||
| The Problem Now Is To Minimize '''''w''''' Subject To The Constraint (3) This Is A | "http://wwwinformationdelightinfo/encyclopedia/entry/quadratic_programming" class="copylinks">Quadratic Programming (QP) Optimization problem |
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