3. (25 points) Consider the following two training points Consider a feature representation (x)-[LV2z,z2 (which is equivalent to using a second order polynomial kernel). Recall the optimal margin classifier in the new 3d feature space min lwl2 Let w* and b* denote the solution of the above optimization problem (a) (5 points)Write down a vector w that is parallel to the optimal vector w*. Hint: for a decision boundary specified by w and b, w is perpendicular to the decision boundary (b) (5 points)What is the value of the geometric margin that is achieved by this w*? Hint 1 Recall the definition of geometric margin Hint 2: Think about the geometry of the point:s in feature space, and the vector between them (c) (5 points)Solve for w*, using the fact that the margin is equal to 1/1lw 12 (d) (5 points)Solve for b* using your value for w* and equation(2). Hint: the inequalities in (2) will be tight (i.e., becomes equalities since the solution to (2) will be on the boundary of the constraint set (5 points) Write down the form of the discriminant function f(x) = w*(x)+b". Plot the 2 points (i),yl1)) and (r2,y/2)) in the dataset, along with f(x) in a 2d plot. You may generate this plot by hand, or using a computational tool like Matlab. (e) 3. (25 points) Consider the following two training points Consider a feature representation (x)-[LV2z,z2 (which is equivalent to using a second order polynomial kernel). Recall the optimal margin classifier in the new 3d feature space min lwl2 Let w* and b* denote the solution of the above optimization problem (a) (5 points)Write down a vector w that is parallel to the optimal vector w*. Hint: for a decision boundary specified by w and b, w is perpendicular to the decision boundary (b) (5 points)What is the value of the geometric margin that is achieved by this w*? Hint 1 Recall the definition of geometric margin Hint 2: Think about the geometry of the point:s in feature space, and the vector between them (c) (5 points)Solve for w*, using the fact that the margin is equal to 1/1lw 12 (d) (5 points)Solve for b* using your value for w* and equation(2). Hint: the inequalities in (2) will be tight (i.e., becomes equalities since the solution to (2) will be on the boundary of the constraint set (5 points) Write down the form of the discriminant function f(x) = w*(x)+b". Plot the 2 points (i),yl1)) and (r2,y/2)) in the dataset, along with f(x) in a 2d plot. You may generate this plot by hand, or using a computational tool like Matlab. (e)
