1. The Relation between SVD and PCA: Use the SVD theorem (Corollary C.6) for providing an alternative proof of Theorem 23.2.

2. Random Projections Preserve Inner Products: The Johnson-Lindenstrauss lemma tells us that a random projection preserves distances between a finite set of vectors. In this exercise you need to prove that if the set of vectors are within the unit ball, then not only are the distances between any two vectors preserved, but the inner product is also preserved. Let Q be a finite set of vectors in Rd and assume that for every x ∈ Q we havex  ≤ 1.

1. Let δ ∈ (0, 1) and n be an integer such that _ = + 6log(|Q|2/δ) N ≤ 3.

Prove that with probability of at least 1− δ over a choice of a random matrix W ∈ Rn,d , where each element of W is independently distributed according to N(0,1/n), we have |_Wu,Wv_−_u,v_| ≤ _ for every u,v ∈ Q.

Hint: Use JL to bound both          W(u+v) u+v        and       W(u−v) u−v        .