Acrobat Reader File Edit View Sign Window Help B Q @ Thu Oct 13 11:01 PM .. . 501 HW 1.pdf Home Tools 501 HW 1.pdf X ? Sign In T 1 /2 O + 75% 2) Given a D x D square matrix B, we frequently want to factor or express B in terms of other matrices, for instance maybe taking the square root of B. In this problem, for a given matrix B, we want to find a (potentially Search 'Redact' non-square, non-symmetric) matrix A such that B = ATA. - We can generate such a B by randomly generating a k x D matrix A, with each element selected as a standard normal random variable (mean 0, variance 1), and computing B = A" A. From B, we would like Export PDF to (potentially) recover A. - Consider the following approach: for any matrix A, define L as Edit PDF L(A) = XX[B - AT Ali,; , (1) 1=1 1=1 Create PDF the sum of the squares of all the differences between the elements of B and A" A. We want to find A to make L as small as possible. Comment - For a given matrix A, let G(A) be the matrix of derivatives so that Gij = 0/0Ajj [L]. Show that G = 4A (ATA - B). Combine Files - We can try to find A to minimize L by implementing gradient descent: for any given A, we can improve it by taking Anew = A - aG(A), (2) Organize Pages v where a is a small positive constant. For sufficiently small o, you'll get that L(Anew)