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Comment by superbatfish | Hacker News Reader

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...

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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.

I think the point of using t-SNE was to suggest that if the structure were legitimate that t-SNE would find at least some of it.

This is both an interesting/fun visualization exercise and a cautionary story. Apparently UMAP has a tendency to render blobs as rings or loops!

Of course they are artifacts -- everything in that picture is an artifact, by definition of how it is produced. But these artifacts are produced under specific conditions, which presents a certain window into the structure of what is being visualized. t-SNE is a different, older, method compared to UMAP, and it over-emphasizes local structure at the expense of global structure.

Example of what could cause a swirly chain in UMAP: if A related to B and B relates to C but A does not relate to C and so on. IMO that's a valid structure to visualize as a swirly chain. If re-run multiple times, of course you will get that chain in different locations and so on. But it is interesting that it is there.

I wonder if we can somehow figure out what those convergence artifacts actually were, and find a way to replicate them for cool digital art effects.

You mean, like the Mandelbrot set?

what are you trying to say?

The structures that emerge from a series of transformations applied to an initial field (be it the natural numbers or the complex set), could be due to the transformations, or intrinsic of the underlying field. The parent comment stated that we're in the former case, i.e. we are seeing artifacts due to the transformation itself and we're not in front of some new properties of the numbers. It implied that this is a bad thing, or at least that's how I read that 'merely'. My (admittedly cryptic) reply was meant to show that similar results are worth attention as well, just like the Mandelbrot set, and should not be quickly dismissed as an unwanted effect.

so, your point is that the mandelbrot set isn't merely an artifact, if i understand you correctly.

Wait... I'm not sure where this conversation is going.

I say that the beauty (or value or worthiness) of the pictures of the Mandelbrot set comes from the transformations we apply to uninteresting complex numbers.

Similarly, the beauty of the pictures in the article may come from some hidden properties of the underlying prime numbers, or from the transformations themselves, and I don't think that either case would be better than the other.

I said this in reply to a comment that seemingly stated "what a pity, these images are 'merely' due to the transformations". I was objecting to the tone of disappointment that I read in that message.

Convergence artifacts are due to the algorithm converging poorly. This is why it's a "pity" the interesting structure you see in the picture has nothing to do with the distribution of numbers and is just a bug in the algorithm that happens to look nice.

So no, it's not like the Mandelbrot set. It's more like if you wrote a script to visualize the Mandelbrot set and created a bug that made part of the visual you created look like there was an interesting structure by accident, then shared your pictures with a wide audience going "look at this interesting structure I found in the mandelbrot set!" and then someone replies on Twitter with "you have bug in line 124 and when I correct it the structure disappears". Which is why it's a pity.

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