13pattern_qp_art

13pattern_qp_art - Pattern Classication and Quadratic...

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Pattern Classification, and Quadratic Problems (Robert M. Freund) March 30, 2004 c 2004 Massachusetts Institute of Technology. 1
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1 Overview Pattern Classification, Linear Classifiers, and Quadratic Optimization Constructing the Dual of CQP The Karush-Kuhn-Tucker Conditions for CQP Insights from Duality and the KKT Conditions Pattern Classification without strict Linear Separation 2 Pattern Classification, Linear Classifiers, and Quadratic Optimization 2.1 The Pattern Classification Problem We are given: points a 1 , . . . , a k n that have property “P” points b 1 , . . . , b m n that do not have property “P” We would like to use these k + m points to develop a linear rule that can be used to predict whether or not other points x might or might not have property P. In particular, we seek a vector v and a scalar β for which: v T a i > β for all i = 1 , . . . , k v T b i < β for all i = 1 , . . . , m We will then use v, β to predict whether or not other points c have property P or not, using the rule: If v T c > β , then we declare that c has property P. If v T c < β , then we declare that c does not have property P. 2
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We therefore seek v, β that defines the hyperplane H v,β := { x | v T x = β } for which: v T a i > β for all i = 1 , . . . , k v T b i < β for all i = 1 , . . . , m This is illustrated in Figure 1. Figure 1: Illustration of the pattern classification problem. 3
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2.2 The Maximal Separation Model We seek v, β that defines the hyperplane H v,β := { x | v T x = β } for which: v T a i > β for all i = 1 , . . . , k v T b i < β for all i = 1 , . . . , m We would like the hyperplane H v,β not only to separate the points with m different properties, but to be as far away from the points a 1 , . . . , a k , b 1 , . . . , b as possible. It is easy to derive via elementary analysis that the distance from the hyperplane H v,β to any point a i is equal to v T a i β . v Similarly, the distance from the hyperplane H v,β to any point b i is equal to T b i β v . v If we normalize the vector v so that v = 1 , then the minimum distance from the hyperplane H v,β to any of the points a 1 , . . . , a k , b 1 , . . . , b m is then: T T b 1 m min v a 1 β, . . . , v T a k β, β v , . . . , β v T b . We therefore would like v and β to satisfy: v = 1, and min v T a 1 β, . . . , v T a k β, β v T b 1 , . . . , β v T b m is maximized. 4
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This yields the following optimization model: PCP : maximize v,β,δ δ s.t. v T a i β δ, i = 1 , . . . , k T b i β v δ, i = 1 , . . . , m v = 1 , v n , β Now notice that PCP is not a convex optimization problem, due to the presence of the constraint v = 1”. 2.3 Convex Reformulation of PCP To obtain a convex optimization problem equivalent to PCP, we perform the following transformation of variables: v β x = , α = . δ δ 1 Then notice that δ = v = x , and so maximizing δ is equivalent to x 1 maximizing x , which is equivalent to minimizing x . This yields the following reformulation of PCP: minimize x,α x s.t.
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