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Re: Leastsquares scaling
Posted:
Nov 13, 2012 12:17 AM


On Nov 11, 10:55 am, Kaba <k...@nowhere.com> wrote: > Hi, > > Let > > R in R^{d times n} > P in R^{d times n}, and > S in R^{d times d}, S symmetric positive semidefinite. > > The problem is to find a matrix S such that the squared Frobenius norm > > E = SP  R^2 > > is minimized. Geometrically, find a scaling which best relates the > paired vector sets P and Q. The E can be rewritten as > > E = tr((SP  R)^T (SP  R)) > = tr(P^T S^2 P)  2tr(P^T SR) + tr(R^T R) > = tr(S^2 PP^T)  2tr(SRP^T) + tr(RR^T). > > Taking the first variation of E, with symmetric variations, > and setting it to zero gives that > > SPP^T + PP^T S = RP^T + PR^T > > holds in the minimum point. One can rearrange this to > > (SPP^T  RP^T)^T = (SPP^T  RP^T), > > which says that SPP^T  RP^T is skewsymmetric. > But I have no idea how to make use of this fact. Anyone? > > http://kaba.hilvi.org
How do you intend to prevent S from having negative eigenvalues? What if R = P ?



