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Manuel

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07 Apr 2014 Uniform Sampling of a Sphere Create an approximately uniform triangular tessellation of a unit sphere Author: Anton Semechko

Hi, I clearly misunderstood the matlab function 'convhulln' here is a nice explanation of it:

http://www.mathworks.com/matlabcentral/answers/9298-pretty-simple-question-regarding-convhulln-now-that-i-have-k

07 Apr 2014 Uniform Sampling of a Sphere Create an approximately uniform triangular tessellation of a unit sphere Author: Anton Semechko

Hi Anton, Thank you for your soon reply. You are right I did not know that result of triangulation grows exponentially with number of dimensions (n), as you nicely illustrated with the random 100 points, so just for n=20 ‘convhulln’ will return an array of doubles with roughly 8.8E+12 elements and to store that I’ll need 65 400 GB of RAM. I only have 125 GB ☺. What is interesting here is that I do not get matlab ‘out of memory’ error but instead ‘The data is degenerate in at least one dimension – ND set of points lying in (N+1)D space’ . The good news is that I don’t really need to triangulate the points (I don’t need to know indices of the points that comprise the facets of the convex hull) even though it could be used to make sense of the data by sampling some areas and comparing them for instance. Well this takes the function ‘convhulln’ out of the game.
But, how about your code? Why are you amazed it worked for 525D? It basically addresses an optimization problem in 3D that can be for sure addressed in higher dimensions too (perhaps using a more sophisticated scheme like Conjugate gradient, etc). Well the thing is that you have done it and shared it, which all of us appreciate, and I am trying to expand its applicability.
Coming back to the hyper sphere in 525D, I don’t know how such a surface looks like ☺ but I believe that 3 points in a hyper dimensional space will define a triangle (just as they do in 2D and in 3D) so approximating the surface of an hypersphere with a set of triangles makes sense to me, therefore I wonder what do you mean by: “… 525 dimensional convex hull is a terribly poor approximation to a hypersphere…”
Once more thank you for your comments and no, I am not kidding!

06 Apr 2014 Uniform Sampling of a Sphere Create an approximately uniform triangular tessellation of a unit sphere Author: Anton Semechko

Hi Anton. Thank you for sharing your code. I wonder whether you have ever expanded your code to deal with higher dimensions. I need to do sampling on the hypersphere surface (525 – 1029 dimensions) and I accomplish that by normalizing the points draw from the Gaussian distribution, which gives a nice uniform distribution of points on the surface of the hipersphere. Nevertheless I need to refine the sampling so to avoid points laying too close to its neighbors and here is where triangular tessellation sounds just perfect.
I have bypass the 3D checking in your code and it seems to work fine for 525D but it fails when calling ‘convhulln’ function so no triangulation is done therefore I can no make sense of the new point distribution (what you call V).
Do you think ‘convhulln’ is not capable of dealing with 525D or your code ( ParticleSampleSphere ) simply can not be extended to higher dimensions?

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