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If an artificial intelligence could open its pipes and vomit onto a newly-allocated two-dimensional array, I imagine its undigested bits would look very much like this.
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.
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...
Earlier quoted context omitted.
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.
It's also very common in general to use this "one-hot encoding" in statistics and machine learning. 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 tha…
Earlier quoted context omitted.
You mean, like the Mandelbrot set?
what are you trying to say?
Earlier quoted context omitted.
It's also very common in general to use this "one-hot encoding" in statistics and machine learning. 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 tha…
That kind of sounds like normalisation in other contexts ("hey let's just say lightspeed is 1, simplifies the equations"), is that a fair analogy to make?
However this encoding technique is a bit more like choosing 1 and 0 to represent Boolean values in C: it's convenient, it's easy to reason about, it's mathematically simpler than any other option, and there's no compelling reason to choose anything else anyway.
For binary variables in particular, you sometimes see people using -1 and 1 instead of 0 and 1, to get symmetry around 0. I think this was mostly only used for encoding the labels/outputs of SVM models, where it's mathematically appealing as representing two sides of a hyperplane.
There are a handful of other schemes for encoding "categorical" or "nominal" data of this kind, used in certain statistical applications such as the design and analysis of experiments. These encoding schemes are called contrasts in the stats literature, because they emphasize the differences (the "constrasts") between categories.
Can someone explain to me like I am 5 ?
Earlier quoted context omitted.
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 ba…