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What do numbers look like?

johnhw.github.io

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Re: What do numbers look like?

#21

Post by teo_zero »

LanceH wrote: post #4
Jtsummers wrote: post #3

Earlier quoted context omitted.

Convert numbers to a vector space represented by their prime factors: 2 => [1 0 0 0...] 3 => [0 1 0 0...] 4 => [1 0 0 0...] And map that high dimensional space back down to two dimensions (using some technique I haven't dug into yet). Colors are assigned by some scheme, later images help to illustrate how the particular clusterings happen like one where primes are rendered in white.

The whole thing seemed elaborately overly-formally-described and that just lessened the value of the cool graph.

I dissent. The list of ingredients of the cake I'm eating doesn't lessen its exquisite taste.

Re: What do numbers look like?

#22

Post by tuukkah »

krick wrote: post #15

I don't get it. So, do all these clusters mean anything at all? It feels like that should be telling us something eerily important about numbers. Granted, I don't really understand UMAP, but still, if it's any good for dimensionality reduction at all (and it appears to be), then clusters are clusters. This seems to have way too distinct structure to be essentially just a weird artifact of UMAP itself. Or is it?

It tells about the factorizations if anything, as that's the input dimensions they are using. If you haven't seen factorization diagrams, it's worth checking them out first: https://mathlesstraveled.com/factorization/

But visualisations can always deceive you into seeing something that's not there, e.g. correlation vs causation.

Re: What do numbers look like?

#25

Post by 2-718-281-828 »

teo_zero wrote: post #23
superbatfish wrote: post #13

According to Dmitry Kobak, some details in these figures are merely convergence artifacts, and no longer produced when using more recent versions of UMAP. https://twitter.com/hippopedoid/status/1318917878364672001?l...

You mean, like the Mandelbrot set?

what are you trying to say?

Re: What do numbers look like?

#26

Post by 2-718-281-828 »

Jtsummers wrote: post #3
qwertyuiop_ wrote: post #2

Can someone explain to me like I am 5 ?

Convert numbers to a vector space represented by their prime factors: 2 => [1 0 0 0...] 3 => [0 1 0 0...] 4 => [1 0 0 0...] And map that high dimensional space back down to two dimensions (using some technique I haven't dug into yet). Colors are assigned by some scheme, later images help to illustrate how the particular clusterings happen like one where primes are rendered in white.

2 and 4 are mapped to the same vector?

Re: What do numbers look like?

#27

Post by krick »

tuukkah wrote: post #22
krick wrote: post #15

I don't get it. So, do all these clusters mean anything at all? It feels like that should be telling us something eerily important about numbers. Granted, I don't really understand UMAP, but still, if it's any good for dimensionality reduction at all (and it appears to be), then clusters are clusters. This seems to have way too distinct structure to be essentially just a weird artifact of UMAP itself. Or is it?

It tells about the factorizations if anything, as that's the input dimensions they are using. If you haven't seen factorization diagrams, it's worth checking them out first: https://mathlesstraveled.com/factorization/ But visualisations can always deceive you into seeing something that's not there, e.g. correlation vs causation.

Uh… what does it have to do with these "factorization diagrams"? Maybe I'm missing something, but I don't even see why are they "worth checking out". As far as I can see following your link, these are just arranging a number of dots into (pre-determined) shapes that are humanly recognizable. I.e., these are literally just some caveman technologies for writing a number before a more convenient (i.e. arabic) number system was invented. If anything, that combination of squares and triangles is less readable way to write 2²×3³×5, and it can be constructed only for "convenient" numbers (well, you can arrange into blocks 23×11×2 and 23×13×2 too, but good luck telling them apart). That's just silly, and tells me absolutely nothing about relationships between numbers.

However, all numbers from 1 to 1 000 000 forming a distinct clusters when being mapped to 2 dimensions with UMAP… I don't know. It might be nothing (like a representation of something trivial, like an observation, that multiples of 100003 are less common than multiples of 3 in the set of first 1 000 000 integers), and all these clusters may just disappear (converge) as we go closer to infinity. But there's definitely something a bit eerie about the possibility of it not being "nothing". Normally, you wouldn't expect any patterns to form like that.

Or, well, it may be more that a "nothing" but less than "interesting" for a mathematician — maybe there actually is some pattern that becomes more visible in this visualization, but it's already well-known among number theorists. I just have no idea, that's why I'm asking. It's just weird to see any clustering at all here.

(And, yeah, BTW, there's no such thing as "correlation vs causation" in number theory.)

Re: What do numbers look like?

#28

Post by sigmoid10 »

superbatfish wrote: post #13

According to Dmitry Kobak, some details in these figures are merely convergence artifacts, and no longer produced when using more recent versions of UMAP. https://twitter.com/hippopedoid/status/1318917878364672001?l...

Beware that the first tweet uses t-SNE, which is an older algorithm that UMAP tries to improve. There's also an image with a newer version of UMAP further down and while the big squiggly line artifacts are reduced, a lot of the structure remains and it looks much less like the random numbers image from the blog or the t-SNE version. Still, I think it's safe to say that any fancy structure here is more likely a result of the algorithm and less of an actual structure in the numbers.

Re: What do numbers look like?

#30

Post by jerpint »

eximius wrote: post #20

I wonder why it was chosen to represent everything, well, not unit vectors, but nothing higher than 1. Why should 2 and 4 both be [1 0 ...] instead of [2 0 ...], etc?

It might have to do with UMAP making dot products and assumptions about the inputs. If everything is 0s and 1s, the vectors will have a normal distribution of magnitudes (more or less). Otherwise the magnitudes will just explode and I don’t think UMAP will work.
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