Problem 3 (Basic Theory Related to the Lasso [20 points]) 3.i Consider the univariate lasso objective function with no bias: Q (B) = 2n (yi - I;B)2 + X/BI Also suppose x is scaled using the formula i = 1, 2, . .., n. Derive a closed form expression for the lasso solution, i.e., show Q(B) is minimized at n Zitiyi - > if B 0 if n Li liyi + A if 3.ii Consider the multivariate lasso objective function with no bias: Q(B) = Q(B1, B2, . .. . Bp) = . (yi - EBjay ) + ZIBil Also suppose that the jth feature x; is scaled using the formula Cij i = 1, 2, ..., n, j = 1, 2, . .., p. Solve the expression aB, 20 = 0 for B;. Your final answer should be where x; is the jth feature, r) is the partial residual, and Sx is the soft thresholding operator. See the lecture slides for further details