Date: Nov 13, 2012 12:17 AM
Author: Ray Koopman
Subject: Re: Least-squares scaling
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 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?

>

> --http://kaba.hilvi.org

How do you intend to prevent S from having negative eigenvalues?

What if R = -P ?