```Date: Nov 16, 2012 11:05 AM
Author: Kaba
Subject: Least-squares using polar decomposition

(I'm sending this again, since the last post did not appear on my news server, nor in Google Groups)Hi,Let P, R in R^{d x n}, A, Q, S in R^{d x d} with Q^T Q = I and S^T = S.The problem is to find A to minimize     |AP - R|,where |.| is the Frobenius norm. It can be shown that the solution is     A = RP^T (PP^T)^{-1}.I want to derive the result differently. First, notice that every A can be decomposed as A = QS. For if A = UDV^T is the singular value decomposition of A, then Q = UV^T and S = VDV^T is such a decomposition. Now I want to solve a new problem, which is to find Q and S to minimize     |QSP - R|.Clearly, I could just compute A and then decompose it as above. But I don't want to do that. Instead, I want to derive the result by optimizing Q and S.If I vary Q (subject to it being orthogonal), I get    PP^T S + SPP^T = Q^T RP^T + PR^T Q.If I vary S (subject to it being symmetric), I get    Q^T RP^T S = SPR^T Q.But then I am stuck. In a previous post this week, concerning least-squares scaling where Q = I, it was noticed that    PP^T S + SPP^T = RP^T + PR^Tis a Lyapunov equation, and can be solved as it is. Therefore, I would except that the solution of this more general problem yields something similar to a Lyapunov equation, perhaps a Sylvester equation (more general), or an algebraic Riccatti equation (even more general):http://en.wikipedia.org/wiki/Sylvester_equationhttp://en.wikipedia.org/wiki/Algebraic_Riccati_equationIt seems to me that first I should be able give an implicit equation for either S or Q alone. It probably is also the case that the QS decomposition (polar decomposition) is not unique, and therefore an additional condition would be needed to make the solution unique.Ideas appreciated. The reason I am interested in this is that I suspect there is a new technique (to me) underlying here.-- http://kaba.hilvi.org
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