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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

by krick 4y ago

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?

tuukkah 4y ago
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/ https://mathlesstraveled.com/factorization/ But visualisations can always deceive you into seeing something that's not there, e.g. correlation vs causation.
- krick 4y ago
  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.)
  - nerdponx 4y ago
    It's the same because both are visualizations of prime factorizations. And if you check out the Twitter thread posted in the comments here, you'll see that the "loops" are probably convergence artifacts (they should just be blobs) and that the clusters seem to correspond to the number of prime factors and the largest prime factor: https://twitter.com/hippopedoid/status/1318917905736716288 https://twitter.com/hippopedoid/status/1318917905736716288, which makes sense because those are the input features to the algorithm.

by krick 4y ago

tuukkah 4y ago

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/ https://mathlesstraveled.com/factorization/ But visualisations can always deceive you into seeing something that's not there, e.g. correlation vs causation.

krick 4y ago
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.)
- nerdponx 4y ago
  It's the same because both are visualizations of prime factorizations. And if you check out the Twitter thread posted in the comments here, you'll see that the "loops" are probably convergence artifacts (they should just be blobs) and that the clusters seem to correspond to the number of prime factors and the largest prime factor: https://twitter.com/hippopedoid/status/1318917905736716288 https://twitter.com/hippopedoid/status/1318917905736716288, which makes sense because those are the input features to the algorithm.