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Given a set of n points, I have to find a subset of given size m<size(n), so that the points in the selected subset are as uniformly distributed as possible across the volume enclosed by the convex hull of set n.

I am solving this problem using Evolutionary Algorithm (EA). The parameters of the problem are the n points and the size m. What is the objective function to be minimized? Each generation, EA finds solutions, where each solution is m points from set n. How to measure which solution is better, meaning which points are more uniformly distributed?

I tried minimizing electrostatic potential ($$E=\frac{1}{2}\sum_i\sum_j\frac{1}{|r_i-r_j|}$$) following the idea of the Thomson problem, however the resulting output is clearly incorrect as the density of the selected points is higher at the edges. I tried squaring values in denominator of given equation for E, but the solutions are still clearly sub-optimal.

What is the objective function I should minimize to get what I want? Note that I want to solve this problem in high dimension space for arbitrary set of points with unknown distribution.

Liisjak
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1 Answers1

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(Disclaimer: I am not an expert on this topic)

You may be interested in finding a set with low discrepancy. From the Wikipedia page :

With these definitions, discrepancy represents the worst-case or maximum point density deviation of a uniform set.

So I guess it makes sense to try to find the set of m (among your n) points minimizing such discrepancy. This looks like a difficult combinatorial problem, plus the discrepancy objective may be expensive to evaluate. The Wikipedia page also mentions alternative, cheaper discrepancy measures :

For instance, L2 discrepancy or modified centered L2 discrepancy are also used intensively to compare the quality of uniform point sets. Both are much easier to calculate for large N and s.

paulduf
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