Problem 4. In class, we have dealt mostly with classication problems. One alternative problem, that is quite frequent in practice, happens when only...

The question is in the plot. It's a vertex optimization problem in the SVM machine learning field.

Problem 4. In class, we have dealt mostly with classification problems. One alternative problem, thatis quite frequent in practice, happens when only data from one class is available (there are no negativeexamples) but the goal is to detect outliers. This problem can be solved in at least two ways usingSVM related ideas. They both imply finding the region of support of the data by finding a minimalboundary surface that includes the data from the class. The first strategy is to find the plane that separates, with largest margin, the training points fromthe origin. It consists of solving the following optimization problem . 1 2 1 "mm (§IIW| +E;€i—p), 1160.1] ngm subject to <@(x,),w> Ep—Ei, i=1,...,n€i>07 i=1,...,n. The second strategy consists of finding the smallest ball that encircles the datapoints. This can bedone by solving the following optimization problem subject to llflmkcl? SR2+€., i=1,...,n§i>0, i=1,...,n. a) Determine the dual problem associated with each of the strategies above as well as the resultingdecision function. b) How would you recover the p parameter in the case of first strategy? Also, in this case, what happenswhen 1/ approaches zero? c) Determine the set of kernels that make the two strategies equivalent. d) In the case of c), compare the decision function with a Parzen density estimate f(x) = figment where g6 is a non—negative function that integrates to one.