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I have two algorithms, both of which create points on the surface of a higher dimensional sphere (say in $\mathbb{R}^n$). Now, I want to check, which algorithm gives a more uniform spread of points over the surface of the sphere. In lower dimensional cases like $\mathbb{R}^2$ and $\mathbb{R}^3$ both perform well but I want to check in higher dimension, say $n \geq 50$. I believe, some kind of clustering algorithm may help but I don't know which one to use in this situation (since the points are on surface and not inside). Is there a way to solve this problem ?

I can generate as many points as required for both the algorithms, if that helps.

user402940
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1 Answers1

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Your question doesn't have many specifics, so my assumption is that you need a way to check how uniform a distribution of points over a sphere is, so as to compare your two algos.

Here are two simple suggestions:

  • Empirically estimate the entropy or uniformity. Essentially, divide the sphere into segments (e.g. in spherical coordinates, but be careful) and count the number of points that fall into each segment. You could then measure the entropy of the histogram (i.e. if $n_i$ points fall into area $i$, and $N$ points were sampled in total, then $p_i = n_i/N$ and the entropy is $\mathbb{H}=-\sum_ip_i\log p_i$. Higher entropy means more uniformity. You could also compute the variance in the set of observed probability values to estimate uniformity, i.e. $\mu_p = \frac{1}{N}\sum_ip_i$, then $V = \frac{1}{N-1}\sum_j (p_j - \mu_p)^2$.

  • Fit a von Mises distribution to the data, and simply check the concentration parameter $\kappa$. As $\kappa\rightarrow 0$, the distribution approaches the uniform one. (See the link for how to do parameter estimation).

user3658307
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