31–40 of 55 posts
This is one of the most beautiful things I have seen in 2022 on the Internet, and there were some very good contenders. This provokes in me an immediate sense of beauty, and I'm compelled to read and understand as much of it as I can. Thanks for sharing this, I'm fascinated and amazed.
Sadly merely artifacts .
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.
ahem .... explain like I am 0.5
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.
2 and 4 are mapped to the same vector?
This is one of the most beautiful things I have seen in 2022 on the Internet, and there were some very good contenders. This provokes in me an immediate sense of beauty, and I'm compelled to read and understand as much of it as I can. Thanks for sharing this, I'm fascinated and amazed.
Sadly merely artifacts .
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.
ahem .... explain like I am 0.5
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…
This is both an interesting/fun visualization exercise and a cautionary story. Apparently UMAP has a tendency to render blobs as rings or loops!
Earlier quoted context omitted.
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 sys…
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.
In many cases using all 2s or all -10s would produce the exact same result in theory, but with more work by the optimizing algorithm, possibly with adverse results as described above.
It's easy to reason about mathematically, too. If the input vector is all 1s and 0s, it's easy to read off the result of multiplying that vector with another vector. Norms of binary vectors and dot products between binary vectors are super-easy, and have a nice correspondence with counting the appearances of elements.
It also corresponds to an array of Boolean values which is conceptually appealing, because that's basically how it's constructed.
With a couple of specific exceptions, there's little reason not to use all 1s.
This is one of the most beautiful things I have seen in 2022 on the Internet, and there were some very good contenders. This provokes in me an immediate sense of beauty, and I'm compelled to read and understand as much of it as I can. Thanks for sharing this, I'm fascinated and amazed.
Sadly merely artifacts .