Problem 3: One approach to applying SVM in settings Where the data classes are not linearly separable is to use Kernels and feature maps to transform the problem space. Let X be a set in RP. A symmetric function K2XXXiHR (symmetric meaning K(E, y) = K (y, 33)) is a positive denite kernel if for any n E Z > 0, and set of points {1:1, . . . , gun} in X, we have K = [Km] = [K($m$j)l is a positive semidenite matrix in Rm\" (we will be covering this in detail in lecture). If this holds, then we can nd a feature map (or function) 9b : X I> V (with d usually larger than p) such that Kt'b', y) = @1503) ' My) (where \"-\" is the dot product in V). The goal is to have the classes linearly separable in feature Space. i. Consider the kernel function (showing it is a kernel is nontrivial here, you can take it as known) Km 9) = (1 + $502 where as, y E R2. Show that (where 3: = (3:1, 552)) the feature map associated with this kernel is (2503) = (13 st/'17 @132, (ti-(Lg, flb'g) E R6. ii. Consider the following data (note that the red circle has radius 4 and is centered at the origin): Clearly, the blue points (class 1) and the green x's (class -1) are not linearly separable in R2. Show that these points are linearly separable in six-dimensional feature space using the feature map o defined above. (Hint: show that the feature map maps the red circle to a hyperplane)