- (0.5 pt) Assume that the true density is L-Lipschitz: Ip* (x) - p*(y)| da ( M). Consider i.i.d. data D= {X1, ..., Xn} with X; E Rd and define their mean Xn and their covariance matrix En = _ _ ( X; - Xn) ( X; - Xn) and an eigenvalue decomposition En = Vdiag( Al ( En), 12( En), ..., Ad ( En) ) VT. The objective is to show that E(k) = Vdiag(Al (En), 12( En), . .., Ak ( En), 0, 0, .. ., O ) VT 3is the best approximation of En of rank k in the sense of the spectral norm (the square root of the largest eigenvalue, denoted by ||M| | for a matrix M): [En -5."| |En - S(k)| for any S(k) of rank k. . (0.5 pt) Give the spectral norms of En, En and En - E.". (0.5 pt) Consider S() a matrix of size d x d and of rank k with eigenvalues. Give the value of A, (S(k)) for any j 2 k + 1 Weyl's inequality (see https://terrytao. wordpress.com/tag/weyl-inequalities/) assures that if A and B are matrices of size d x d with eigenvalues. Then for all i, j 2 1 and i t j - 1 S d Nitj - 1 ( A + B) S Xi (A) + x; ( B). . (0.5 pt) Write the previous inequality with A = En - S(*) and B = S(*). What happens if j 2 k + 1 ? . (1 pt) Use Weyl's theorem and the result of the previous question to show that the spectral norm of En - S() is greater than the one of En - En. 4