1. [written, 4×5 = 20 pts] Kernel machines short answer (Show your work for all the questions!) (a) Show that an SVM that employs the quadratic Kernel function, K(x,y) = x ⊺y, can successfully...




1. [written, 4×5 = 20 pts] Kernel machines short answer (Show your work for all the questions!) (a) Show that an SVM that employs the quadratic Kernel function, K(x,y) = x ⊺y, can successfully classify all the instances in a dataset with below four instances { (x1 = 1, x2 = 1),r = 1  , (x1 = 1, x2 = −1),r = −1  , (x1 = −1, x2 = 1),r = −1  , (x1 = −1, x2 = −1),r = 1  } (b) Consider the following two points in two-dimensional space: { x 1 = (x 1 1 = 0, x 1 2 = 0),r 1 = 1  , x 2 = (x 2 1 = 1, x 2 2 = 1),r 2 = −1  } We can define a line equation as x ⊺w+1 = 0, where w is a vector of length 2. What w vector separates x 1 and x 2 with the largest margin? (c) Consider an SVM with decision boundary defined by the hyperplane w ⊺φ(x)+b = 0 for a two-dimensional (2D) feature space, where φ(x) = (x1, x2, q x 2 1 +x 2 2 ) transforms 2D inputs to 3D. Let w = [−2,−1,0] and b = 1. Assume that we have the following data points in the training set: ⋄ x 1 = (0,0), r 1 = −1 ⋄ x 2 = (−1,0), r 2 = 1 ⋄ x 3 = (3,4), r 3 = 1 What would be the slack values (ξ 1 ,ξ 2 ,ξ 3 ) for these training instances (for fixed (w,b) values)? (d) Consider the following dual model that can be used to identify the support vectors in a Kernel machine model: max Ld = ∑t α t − 1 2 ∑t ∑s α tα s r t r sK(x t ,x s ) s.t. ∑t α t r t = 0, and 0 ≤ α t ≤ C, ∀t Let K(x t ,x s ) = (x t ) ⊺x s 2 = (x t 1 x s 1 +x t 2 x s 2 ) 2 , α t = [2,1,1] and C = 2. (i) What is the corresponding objective value (i.e., Ld) for a training set consisting of {(x 1 ,r 1 ),(x 2 ,r 2 ),(x 3 ,r 3 )} provided in part (c)? (ii) We can calculate the discriminant value as g(x) = ∑t α t r tK(x t ,x). Calculate the corresponding discriminant function values (g(x 1 ),g(x 2 ),g(x 3 )) and the resulting predictions (y 1 , y 2 , y 3 ) for the data instances provided in part (c) (Note that this can be used to calculate training set error when {(x 1 ,r 1 ),(x 2 ,r 2 ),(x 3 ,r 3 )} are in the training set ?
Feb 07, 2023
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