it is PCA (principal component analysis) problem

11. Principal Component Analysis [11 points] Suppose that you are given a dataset of NV samples, i.e., x ( R for i = 1, 2, ..., N. We say that the data is centered if the mean of samples is equal to 0, i.e., NE,_ x() = 0. For a given unit direction u such that |ull2 = 1, we denote by Pu(x) the Euclidean projection of x on u. Recall that projection is given by Pu(x) = u xu (a) [3 points] Mean of data after projecting on u: Show that the projected data with samples Po,(x()) in any unit direction u is still centered. That is show, ~ [Pu(x() = 0. (b) [4 points] Marimum variance: Recall that the first principle component u, is given by the largest eigenvector of the sample covariance (eigenvector associated to the largest eigenvalue). That is, N u. = argmaxu u: |luz=1 i=1 Show that the unit direction u that maximizes the variance of the projected data corresponds to the first principle component of the data. That is, show 1. = argmax \\ RX"-4 PIXGOR u: |u2=1 =1 (c) [4 points] Minimum error: Show that the unit direction u that minimizes the mean squared error between projected data points Pu(x)) and the original data points x) corresponds to the first principal component u.. That is show, U. = argmin >x()-P.(x2)13- (1) u : u|2=1