This is the first of two posts based on a talk given at Dmitriy Bilyk at our probability seminar. I don’t usually go to probability seminar, but have enjoyed Dmitriy’s talks in the past, and I didn’t regret it this time either :)
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Unsatisfactory Methods
Here is how not to pick a point uniformly on the sphere: don’t pick its polar and azimuthal angles uniformly. This doesn’t work because it will incorrectly bias things toward the poles.
The problem is that this cylindrical trick only works on the 2-sphere. There is no natural notion of “cylindrical coordinates” for higher-dimensional spheres, because how many coordinates do you take linearly and how many do you take as angles?
Bilyk did not say this, but presumably no choice you can make, or at least no consistent choices that you can make for all $n$, such that you get a uniform distribution— otherwise it really would have been a short talk :P
What Bilyk did say is that there are several ways to choose points uniformly from a sphere, but “there are very few deterministic methods”. But before we can tackle these methods, we actually need to answer a more fundamental question: what does it mean to choose deterministically choose points uniformly? Generally “choosing uniformly” means sampling from a constant random variable, but that X is clearly not available to us here.
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Riesz Energy
One way to go might be to minimize the Riesz energy. Whenever we have a collection of points $Z=\{z_1,\cdots z_N\}$, can write
We see that if the $z_k$ are very close to each other, this energy is going to be large; a set $Z$ that makes the energy small will be one whose points are generally “as far from each other as possible”. Since it’s a sphere, you can’t get too far away, and so there’s an interesting optimization to be done here.
So this seems nice, but there’s a problem. It turns out that minimizing the Riesz energy is just, like, stupendously hard. The best exact result we have was given in 2010, when Schwartz proved that $E_1$ is minimized for $N=5$ (!) by the triangular bipyramid.
To give some indication about why the problem is hard: the triangular bipyramid is not the minimizer for the Riesz energy with $N=5$ for all $s$. It is suspected that it minimizes it for all sufficiently small $s$; but one thing we know for sure that when $s$ gets large enough, the square pyramid is better.
Conjecture. There exists a critical value $s’$ such that for all $1\leq s<s’$, the triangular bipyramid minimizes $E_s$, and for all $s’<s$, the square pyramid minimizes $E_s$.
This conjecture is wide open: we don’t even know that the square pyramid minimizes $E_s$ for any value of $s$!
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Discrepancy and a Theorem of Beck
Another, rather different way to measure the uniform-ness of a set $Z$ is by computing its discrepancy. The formal definition of discrepancy is really a lot scarier than necessary, so I won’t write it here. The idea is that if you pick any (measurable) set $S$, you can either
calculate the measure of the $S$, or
count how many of the points $z_i$ live inside $S$
Most of the time, these two numbers are going to be different, and the discrepancy $D(Z,S)$ is the difference between the numbers, divided by the number of points in $Z$. But this is not the most useful number because we have this extra data $S$ hanging around; the better idea is to let the discrepancy be the maximum of $D(Z,S)$ over all $S$ (technically the supremum).
Intuitively speaking, that if you were to have a $Z$ for which the discrepancy in the latter sense were small, then $Z$ looks “uniformly distributed”, even if $Z$ is deterministic.
However, measurable sets can look pretty wacky, and so in order to let geometry reign over set theory, it often helps to be a little more refined. Given a collection of sets $\mathcal S$, we say that the discrepancy $D(Z,\mathcal S)$ is the supremum of $D(Z,S)$ over all $S\in\mathcal S$. So it’s basically the same thing as what we did above, except that instead of doing all measurable sets, we only do the ones in the collection.
Figuring out optimal discrepancies is also not very easy, but people have over the years figured out strategies for determining asymptotic bounds. And even that tends to be pretty tough. For instance, if we’re considering things on the sphere, it may seem reasonable to look at $\mathcal S$ the collection of spherical caps:
What is the best available discrepancy in this setting? The answer, morally speaking, is that it’s “close to $1/\sqrt{N}$”, but it has a small dimensional correction:
Theorem (Beck 1984). For any positive integer $N$, there exist positive constants $C_0$ and $C_1$ such that
So you can always find a set $Z$ that does a little better, asymptotically, than $1/\sqrt{N}$; what exactly “a little” means depends on how high-dimensional your sphere is.
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In the next post, we talk about more recent developments, including a third notion of uniform distribution which will bring us all the way to Bilyk’s work in the present day.
Isn’t the new Pixar short “The Blue Umbrella” just another re-telling of this story as shown in the classic disney short “Johnny Fedora & Alice Bluebonnet"?
…So I made a comic based on the end of this fic I wrote, but all you really need to know is that Dazai’s been pining about Chuuya’s hands since they defeated Randou, and he deals with this in a Very Normal Person way by holding Chuuya’s hand whenever he’s passed out post-Corruption use. This part takes place during that one soukoku scene in Dead Apple in the fog.
… also there are two parts to this, each with nine pages, and I’m putting most of the pages under read mores because otherwise this is going to be hellish to scroll past
