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peculiar min-max question
Posted:
Aug 17, 2011 7:00 AM
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Hallo:
In research work on a certain statistical problem the following min/max
problem and the question of a fast algorithm came up:
Given n vectors x_1,...,x_n in R^d and denote by \|\theta\| the
Euclidian norm of \theta\in R^d. I would like to efficiently compute
(or at least approximate)
V_n=min_{\|\theta\|=1} \max_{1\leq i\leq n}|<x_i,\theta>|
where <x_i,\theta> denotes the ususal inner-product.
The brute-force approach works, sort of.... Here is what I tried:
Sample m independend and uniformly distributed \theta_j on the sphere,
compute the m\times n matrix of |<x_i,\theta_j>| and then find the
min/max of that matrix. This is easily done in MATLAB. However, n and
the dimension d (to have m small enough) have to be reasonable small...
Therefore I am looking for a different approach. For me it look's like a
problem in convex geometry or optimization....
Since I am not an expert in these fields of mathematics (me working
mostly in probability theory) I would appreciate any suggestions or remarks.
Many thanks!
Peter
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