Date: Nov 11, 2012 1:55 PM
Author: Kaba
Subject: Least-squares scaling



R in R^{d times n}
P in R^{d times n}, and
S in R^{d times d}, S symmetric positive semi-definite.

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 skew-symmetric. But I have no idea how
to make use of this fact. Anyone?