Support Vector Machines II. CAP 5610: Machine Learning Instructor: Guo-Jun QI
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1 Support Vector Machines II CAP 5610: Machine Learning Instructor: Guo-Jun QI 1
2 Outline Linear SVM hard margin Linear SVM soft margin Non-linear SVM Application
3 Linear Support Vector Machine An optimization problem Objective: Maximizing the margin of the linear classifier 3
4 Classifier Margin x Parameters f y est denotes +1 denotes -1 f(x,w,b) = sign(w x + b) Define the margin of a linear classifier as the width that the boundary could be move by before hitting a data point on two sides.
5 Basics to SVM math w/ w : Perpendicular to line wx+b=0 Unit length Margin of two parallel lines is w/ w w x x 1 w w x 1 where wx wx 1 b b 1 1 w( x x ) 1 w/ w x 5
6 Linear SVM Mathematically Goal: 1) Correctly classify all training data ) Maximize the Margin same as minimize if y i = +1 if y i = -1 for all i M 1 w w t w We can formulate a Quadratic Optimization Problem and solve for w and b Minimize subject to wx i b 1 wx i b 1 y i ( wx b) 1 ( w) y i i 1 t w w ( wx b) 1 i i
7 Linear Support Vector Machine An optimization problem Objective: Maximizing the margin of the linear classifier Constraints: Correctly separating two classes 7
8 Constraints denotes +1 denotes -1 Hard Margin: So far, all data points are classified correctly No misclassification error wx i b 1 wx i b 1 ( wx b) 1 y i i if y i = +1 if y i = -1 for all i
9 Linear SVM Mathematically We can formulate a Quadratic Optimization Problem and solve for w and b 1 t ( w) w w y i ( wx b) 1 i i Dual problem: Use Lagrange multiplier. α i is associated with every constraint : y ( wx b) 1 i i Find α 1 α N such that Q(α) =Σα i - ½ΣΣα i α j y i y j x it x j is maximized and (1) Σα i y i = 0 () α i 0 for all α i
10 New Challenges: Nonlinear Separable Case denotes +1 denotes -1 Hard Margin: So far, all data points are classified correctly - No misclassification error What if the training set is not linearly separable?
11 x 1 x ξ Soft Margin Classification Previous constraints y i (w T x i + b) 1 Slack variables ξ i to allow misclassification: y i (w T x i + b) 1- ξ i ξ i 0 ξ 1 ξ 3 What should our quadratic optimization criterion be? We expect ξ i to be small. x 3 Φ(w) =½ w T w + CΣξ i Move the support vector plane so that misclassified data will be correctly classified.
12 Hard Margin v.s. Soft Margin The old formulation: Find w and b such that Φ(w) =½ w T w is minimized and for all {(x i,y i )} y i (w T x i + b) 1 The new formulation incorporating slack variables: Find w and b such that Φ(w) =½ w T w + CΣξ i is minimized and for all {(x i,y i )} y i (w T x i + b) 1- ξ i and ξ i 0 for all i Similar solution can be obtained to that of hard margin Parameter C can be viewed as a way to control overfitting.
13 Outline Linear SVM hard margin Linear SVM soft margin Non-linear SVM Application 13
14 XOR problem x x 1 x x 1 x 1 XOR data are not linearly separable Mapping (x 1, x ) to (x 1, x 1 x ) 14
15 Non-linear SVMs: Feature spaces General idea: the original input space can always be mapped to some higher-dimensional feature space where the training set is linearly separable: Φ: x φ(x)
16 Non-linear SVMs If every data point is mapped into high-dimensional space via some transformation Φ: x φ(x), optimization problem is similar: Find α 1 α N such that Q(α) =Σα i - ½ΣΣα i α j y i y j φ(x i ) T φ(x j ) is maximized (1) Σα i y i = 0 () α i 0 for all α i Classifying function is: f(x) = Σα i y i φ(x i ) T φ(x) + b But relies on inner product φ(x i ) T φ(x)
17 The Kernel Trick SVM relies on Linear: K(x i,x j )=x it x j Non-linear: K(x i,x j )= φ(x i ) T φ(x j ) Feature mapping is time-consuming. Use a kernel function that directly obtains the value of inner product Feature mapping φ is not necessary in this case. Example: -dimensional vectors x=[x 1 x ]; let K(x i,x j )=(1 + x it x j ), It is inner product of φ(x) = [1 x 1 x 1 x x x 1 x ] Verify: K(x i,x j )=(1+x it x j ) = 1+ x i1 x j1 + x i1 x j1 x i x j + x i x j + x i1 x j1 + x i x j = [1 x i1 x i1 x i x i x i1 x i ] T [1 x j1 x j1 x j x j x j1 x j ] = φ(x i ) T φ(x j ),
18 Examples of Kernel Functions Linear: K(x i,x j )= x i T x j Polynomial of power p: K(x i,x j )= (1+ x i T x j ) p Gaussian (radial-basis function network): xi x K( xi, x j) exp( j ) Sigmoid: K(x i,x j )= tanh(β 0 x i T x j + β 1 )
19 Non-linear SVMs Mathematically Dual problem formulation: Find α 1 α N such that Q(α) =Σα i - ½ΣΣα i α j y i y j K(x i, x j ) is maximized and (1) Σα i y i = 0 () α i 0 for all α i The solution is: f(x) = Σα i y i K(x i, x j )+ b Optimization techniques for finding α i s remain the same!
20 Nonlinear SVM - Overview The feature is mapped to a high dimensional space where training data are separable. Inner product is computed by kernel function. Optimization problem is similar to linear SVM
21 Outline Linear SVM hard margin Linear SVM soft margin Non-linear SVM Application gender recognition By Baback Moghaddam and Ming-Hsuan Yang 1
22 Gender recognition Application: Adaptive advertisement Collection of demographic information in shopping mall
23 Recognition rate 1044 male image, 711 female 4/5 for training, 1/5 for testing Higher error rate in classifying female 3
24 Comparison 4
25 Support face Each pair is closest in the projected high dimensional space 5
26 Resources References Vladimir Vapnik: The Nature of Statistical Learning Theory. Springer-Verlag, Christopher J. C. Burges: A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998 Bernhard Scholkopf and A. J. Smola: Learning with Kernels. 00 A useful website: Software: LIBSVM: SVMLight: svmlight.joachims.org/ 6
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