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Topic: Least-squares scaling
Replies: 4   Last Post: Nov 13, 2012 6:57 PM

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Ray Koopman

Posts: 3,383
Registered: 12/7/04
Re: Least-squares scaling
Posted: Nov 13, 2012 12:17 AM
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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 ?



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