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series5 - Exercises Machine Learning Laboratory Dept of...

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Machine Learning Laboratory Dept. of Computer Science, ETH Z¨urich Prof. Dr. Joachim M. Buhmann Web http://ml2.inf.ethz.ch/courses/ml/ Email questions to: Dmitry Laptev [email protected] Exercises Machine Learning AS 2012 Series 5, Nov 21th, 2012 (Adaboost) Problem 1 (Combining Kernels): Recall the definition of positive semidefiniteness. eigenvalues Show that positive semidefinite matrices have non-negative eigenvalues. In this exercise, we explore several common ways to combine known kernels so that we obtain new kernels. For some input data x, x 0 ∈ X , let k 1 ( x, x 0 ) and k 2 ( x, x 0 ) be kernels over X × X → R . Most of the following proofs can be done by giving the corresponding mapping Φ to feature space. scaling For a positive real number a , show that k ( x, x 0 ) := ak 1 ( x, x 0 ) is a kernel. sum Show that k ( x, x 0 ) := k 1 ( x, x 0 ) + k 2 ( x, x 0 ) is a kernel. linear combination For real positive numbers a and b , show using the results before that k ( x, x 0 ) := ak 1 ( x, x 0 )+ bk 2 ( x, x 0 ) is a kernel. product Show that k ( x, x 0 ) := k 1 ( x, x 0 ) k 2 ( x, x 0 ) is a kernel. exponentiated Show that for a positive integer p , k ( x, x 0 ) := k 1 ( x, x 0 ) p is a kernel metric For x, x 0 R d , and for a positive definite matrix M , show that k ( x, x 0 ) := x > Mx 0 is a kernel. Problem 2 (Deriving the SVM Dual): Recall the soft margin C -SVM: minimize w,b,ξ 1 2 k w k 2 + C n X i =1 ξ i (1) subject to y i ( h w, x i i + b ) 1 - ξ i for all i = 1 , . . . , n. (2) ξ i 0 for all i = 1 , . . . , n (3) 1. Write down the Lagrangian, using α i as the Lagrange multiplier corresponding to constraint (2) and γ i as the Lagrange multiplier corresponding to constraint (3).
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