Chap4-Part1 - Linear Models for Classification Discriminant...

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1 Linear Models for Classification Discriminant Functions Sargur N. Srihari University at Buffalo, State University of New York USA
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Topics Linear Discriminant Functions – Definition (2-class), Geometry – Generalization to K > 2 classes Methods to learn parameters 1. Least Squares Classification 2. Fisher’s Linear Discriminant 3. Perceptrons 2 Machine Learning Srihari
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Discriminant Function Assigns input vector x to one of K classes denoted by C k Restrict attention to linear discriminants – Decision surfaces are hyperplanes First consider K = 2 , and then extend to K > 2 3 Machine Learning Srihari
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Geometry of Linear Discriminant Functions: Two-class linear disc fn: y (x) = w T x + w 0 w is wt vector and w 0 is bias Assign x to C 1 if y (x) 0 else C 2 Defines boundary as y (x) = 0 • w determines orientation of surface since w T (x A -x B )=0, w is orthogonal to every vector on surface w 0 sets distance of origin to surface w T x/||w|| = w 0 /||w|| y (x) gives signed measure of perpendicular distance r of point x to decision surface , r =y( x )/|| w || With dummy input x 0 =1 and ω =( w 0 ,w) then y (x) = ω T x – passes thru origin in augmented D+1 dim. space
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Multiple Classes with 2-class classifiers By using several 2-class classifiers 5 Build a K class discriminant Use K 1 classifiers, each solve a two-class problem Alternative is K(K 1)/2 binary discriminant functions, one for every pair One-versus-the-rest One-versus-one Both result in ambiguous regions of input space Machine Learning Srihari
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Multiple Classes with K discriminants 6 Consider a single K class discriminant of the form y k (x) = w T k x + w k0 Assign a point x to class C k if y k (x) > y j (x) for all j k Decision boundary between class C k and C j is given by y k (x) = y j (x) This corresponds to D 1 dimensional hyperplane defined by – (w k w j ) T x + ( w k 0 w j0 ) = 0 Same form as the decision boundary for 2-class case w T x + w 0 =0 Decision regions of such a discriminant are always singly connected and convex Proof follows Machine Learning Srihari
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Convexity of Decision Regions (Proof) 7 Thus R k is singly-connected and convex ˆ x = λ x A + (1 λ ) x B Consider two points x A and x B both in decision region R k Any point on line connecting x A and x B can be expressed as From linearity of discriminant functions y k (x) = w T k x + w k0 y k ( ˆ x) = λ y k (x A ) + (1 λ ) y k (x B ) Because x A and x B lie inside R k it follows that y k (x A ) > y j (x A ) and y k (x B ) > y j (x B ) for all j k.
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