JL Lemma (norm-preserving version): Given a set S ofn vectors in d-dimensional space, there exists a random k X d matrix H where k = 0(log 812), such that for any x E S (1 )|le|: 3 \"Fix\": 3 (1+ e)||x||: with probability at least 1 i. Each entry of the matrix H is an independent Gaussian. n3 In this problem, we are going to consider a different random matrix as follows. For each i E {1 , , d} we pick uniformly random number hi 6 {1 , ..., k}. We then set thi = :1 for each i E {1 , , n} (the sign is chosen uniformly at random from {1, 1}), and all other entries of H are set to 0. Show that for this H, ||l'Ix| | is an unbiased estimator 01' ||x||