Fantastic video from @SciShow about our work that turns any shape into fair dice: piped.video/-gp7AbYD9NI?si=4pA6… Get all the details on Hossein Baktash’s page here: hbaktash.github.io/

Cool. They don’t have to cite it: I released it under CC0 1.0 Universal: creativecommons.org/publicdo… But they were nice enough to cite it anyway. 😁

Here you go! This is a single human hair. If it lands on its end, you win!

“Fair dice” might make you think of perfect cubes with equal frequencies (say, 1/6 on all sides) 🎲 But “fair” really just means you get the frequencies you expect (say, 1/4, 1/4 & 1/2) We can now design fair dice with any frequencies—and any shape! 🐉 hbaktash.github.io/projects/…

I got tired of mashing together tools to write long threads with 𝐫𝐢𝐜𝐡 𝑓𝑜𝑟𝑚𝑎𝑡𝑡𝑖𝑛𝑔 and ℳα†ℏ—so I wrote La𝑇𝑤𝑒𝑒𝑡! It converts Markdown and LaTeX to Unicode that can be used in “tweets”, and automatically splits long threads. Try it out! keenancrane.github.io/LaTwee…

Now please recognize these surface isotopy classes. (…not that any human can do it either.)

Correct. Given a collection of data samples (xᵢ,yᵢ), you might say that a function f is 𝐢𝐧𝐭𝐞𝐫𝐩𝐨𝐥𝐚𝐭𝐢𝐧𝐠 if f(xᵢ) = yᵢ for all i. The definition of an 𝐚𝐩𝐩𝐫𝐨𝐱𝐢𝐦𝐚𝐭𝐢𝐧𝐠 function is not as crisp, but you might say that, e.g., it minimizes error among some class of functions. E.g., the “best-approximating” polynomial of a fixed degree might minimize the sum of squared differences ∑ᵢ ‖f(xᵢ)−yᵢ‖² among all polynomials of that degree.

Forget ReLU: if k < n, the map cannot possibly be injective. Uncountably many points in the data space get mapped to the same point in latent space. This diagram captures that idea: the composition of the encoder and the decoder in fact defines a foliation of the data space.

Likewise, here's a simpler “implementation” diagram, that still retains the most important idea of an *auto*-encoder, namely, that you're comparing the output against *itself*.

Personally, I find both of these diagrams a little bit crowded—here's a simpler “representation” diagram, with fewer annotations (that might anyway be better explained in accompanying text).