4.) Consider first the following binary training data: Class +1: (5,1),(5, -1),(6,1),(6,-1) Class - 1: (1,0).(0,1),(0, -1),(-1,0) a) Plot the points and draw the optimal separating line by inspection (such that y = - !) w b) Write the primal (optimization problem in Eq. 20 in uploaded SVM chapter) Support Vector Machine (SVM) problem corresponding to this instance (not the general formulation!). By indicating the support vectors, give the optimal solutions to the primal SVD problem, w =?, b =? (Hint: note that the inequality constraint in Eq. 20 turns to equality for support vectors). c) Express the constraint of primal SVM problem with respect to b and dual SVM parameters ai's. (Hint: substitute Eq. 22 into the constraint of Eq. 20). Find a's and b from the obtained constraint expression (Hint: use also Eq. 23, set a; = 0 for a non-support vector and note that the inequality constraint turns to equality for support vectors). Write the separating hyperplane in terms of the support vectors (using Eq. 27). Now consider the following data: Class +1: (2,2),(2,-2).(-2,-2),(-2,2) Class -1: (1,1),(1,-1),(-1,-1),(-1,1) d) Plot the points. Are the classes linearly separable? e) Consider the mapping = {14- [[4 x2 + x4 xz1 4 - X2 + 1x2 x21]" , if x + x > 4 [x X21" otherwise Plot the transformed data locations = [0, 0,1" with respect to 0,-axis and 02-axis. Are they linearly separable? f) Repeat (b) and (c) to obtain the optimal separating hyperplane w"o+b = 0 explicitly for this problem instance and use it to classify the point (4,5) in the original (non-transformed space). 8) Consider the following new mapping Q = [x x2 (x + x - 5)/3)" Are the transformed data locations = [0, 0,1" linearly separable? h) Repeat (b) and (c) to obtain the optimal separating hyperplane w"o+b = 0 explicitly for this problem instance and use it to classify the point (4,5) in the original (non-transformed space). 4.) Consider first the following binary training data: Class +1: (5,1),(5, -1),(6,1),(6,-1) Class - 1: (1,0).(0,1),(0, -1),(-1,0) a) Plot the points and draw the optimal separating line by inspection (such that y = - !) w b) Write the primal (optimization problem in Eq. 20 in uploaded SVM chapter) Support Vector Machine (SVM) problem corresponding to this instance (not the general formulation!). By indicating the support vectors, give the optimal solutions to the primal SVD problem, w =?, b =? (Hint: note that the inequality constraint in Eq. 20 turns to equality for support vectors). c) Express the constraint of primal SVM problem with respect to b and dual SVM parameters ai's. (Hint: substitute Eq. 22 into the constraint of Eq. 20). Find a's and b from the obtained constraint expression (Hint: use also Eq. 23, set a; = 0 for a non-support vector and note that the inequality constraint turns to equality for support vectors). Write the separating hyperplane in terms of the support vectors (using Eq. 27). Now consider the following data: Class +1: (2,2),(2,-2).(-2,-2),(-2,2) Class -1: (1,1),(1,-1),(-1,-1),(-1,1) d) Plot the points. Are the classes linearly separable? e) Consider the mapping = {14- [[4 x2 + x4 xz1 4 - X2 + 1x2 x21]" , if x + x > 4 [x X21" otherwise Plot the transformed data locations = [0, 0,1" with respect to 0,-axis and 02-axis. Are they linearly separable? f) Repeat (b) and (c) to obtain the optimal separating hyperplane w"o+b = 0 explicitly for this problem instance and use it to classify the point (4,5) in the original (non-transformed space). 8) Consider the following new mapping Q = [x x2 (x + x - 5)/3)" Are the transformed data locations = [0, 0,1" linearly separable? h) Repeat (b) and (c) to obtain the optimal separating hyperplane w"o+b = 0 explicitly for this problem instance and use it to classify the point (4,5) in the original (non-transformed space)
