Re: Dimensionality Reduction to a Circle


"Warren" <suagw@xxxxxxxxx> wrote in message
news:13027664.1114887904348.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
> Hi Rusty,
>
> Thank you very much for the suggestion. But Multidimensional Scaling is a
> Euclidean metric-preserving method. My problem is trying to preserve the
> arc distances or angles, so I am afraid classic MS doesn't work here. Is
> there any angle-preserving algorithm?

After further thought, if the circle has the same diam as the sphere, then
aren't both arc lengths and Euclidian distances preserved. Assuming arc
length is alway measured the shortest way round the circle that is. Which
is probably what I had in mind, perhaps not very clearly, for the first
reply. Iff this is true, Euclidian metrics can be used. MDS then
reduces to a matrix eigenvalue problem, which is what the molecular
reconstruction folk do eg
http://www.public.iastate.edu/~qfdong/publications/JGO_exact.pdf
and this makes the programming problem fairly easy.
If the diams are not the same then it could get more complicated, though MDS
can work in arbitrary metrics as someone else has posted and only needs to
preserve the sorted sizes of the distances, however they are measured.
Though there are some results in the Davies and Coxon MDS book to the effect
that if Euclidean metrics are used they will get much the same results as
any other distance metric.

rusty




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