Happy to answer any questions.

FWIW, a friend of mine (and not "a friend of mine") wrote this minimal diffusion awhile back that I've found useful that's a bit more "full" w.r.t DDPM. More dropping here since I saw others excited to see code and it could provide good synergy here: https://github.com/VSehwag/minimal-diffusion/

For later topics I think there's a lack on discussions about things like score modeling, the difficulties of understanding latent representations (so many misunderstand the curse of dimensionality), and Schrodinger Bridges. There's so much lol

The better advice I can give is to listen to anyone complaining. It's hard because you have to try to see beyond their words and find their intent (I think this also makes them more digestible and less emotionally draining/hurtful). Especially consider a novice isn't going to have to right words to correctly express their difficulties parsing and it may just come out looking like anger when it's frustration.

Just keep doing your best :)

As someone who's spent a little time experimenting with sampling procedures on stable diffusion, I was also hoping for a comparison to DDIM in terms of convergence time/steps. Is there a relationship between momentum, convergence and error? (i.e, your momentum sampler at 16 steps is ~equivalent to ddim at 20 steps ± %err_term)

thanks for the excellent post

That goes into a lot more mathematical detail but is also accompanied by a minimal, very easy to understand

i'd love an extension of this for the diffusion transformer, which drives Sora and other videogen models. maybe combine this post with https://jaykmody.com/blog/gpt-from-scratch/ and make the "diffusion transformer from scratch" intro

On the other hand, if you want to really dig in, I'd suggest reading into works by Kingma, Gao, Ricky Tian Qi Chen, and honestly, any of Max Welling's students (Tomczak (was post doc), Hoogeboom, etc), and of course the unsung hero Aapo Hyvärinen. Here's a taste at a Kingma & Gao work that's on the lighter side but relevant to the SD3 paper. The unfortunate part is that there's a lot of reliance on knowing and understanding prior works which make these less approachable, but honestly this is a bit difficult to call a meaningful critique (it's research, not educational work aimed at public).

https://arxiv.org/abs/2303.00848

https://news.ycombinator.com/item?id=39665427

It would take quite a long time but I could see this slowly happening for analogous reasons to how the printing press favored alphabetic languages over pictorial ones. Computers teach us to read, write, and express linguistically and are now thanks to LLMs capable of working much more fluently with language. Squiggles and lots of Greek letters seem like an anachronism.

My best guess is never.

The problem is that math is a language itself. In fact, I'd more accurately call it a family of languages. People often confuse it for an inherent thing because most of the time we use it in applied ways for subjects like physics, and the sciences. But clearly those people have never taken abstract algebra or developed their own toy algebra (or group, ring (or rung), or field). You might say they have an _ideal_istic notion of mathematics.

That said, I am absolutely sympathetic to the complexity of mathematics and there is so much that is difficult to grasp. What's worse, is that when this abstract gobbledy goop finally clicks it's so abundantly obvious that it deceives you into thinking you've known it all along. I think this contributes to the difficulty of teaching the subject because those qualified have forgotten the struggles they themselves have had (to be fair, human psychology plays a big role here too because if we only remember the struggles we'd have difficulty finding the passion and beauty).

But I think our best hope is to find a unified mathematics. The obsession people have with things like set theory and category theory is that you can see this interconnectedness. You get this deep feeling that you're looking at the same things manifesting in different ways. Remember that a lot of mathematics is converting your system into another system (we might call this an isomorphic mapping) such that your problem is easier to solve in the other system (think about how we might want to go from Cartesian coordinates to polar, but now abstract that concept out more than your friend that took a few tabs of acid). You zoom out move around, and zoom back in, and maybe a few more times (but you have to keep track of your path).

> Squiggles and lots of Greek letters seem like an anachronism.

Do you have another suggestion? Because I don't see how code, English, or literally any other thing that can be written down is anything more meaningful than "a bunch of squiggles." There's only so many simple, and importantly, distinguishable squiggles that we can write down. Luckily we have a tool to exactly determine this ;) The shitty part of mathematics is also the best part, in that you have to embrace the abstractness of it all. But of course human brains weren't designed for this. (Yes, there are some of us trying to figure out how to get machines to think with that extreme level of abstractedness at its core. No, this is not how computers already "think.")

So yeah, I empathize with you. Quite a lot. There's so much more that I want to know about this thing too. But it is often seemingly impenetrable. All I can tell you is that it just looks that way and isn't. Everyone has the capacity but finding the right way up this mountain (that I'm nowhere near Terry, and he's nowhere near the summit) can be just as hard as the climb itself. The best way is persistence though. Just remember that you don't play video games because they are easy. Sometimes you gotta just trick yourself into pushing through. And some of the best advice I can give is revisiting old topics. So much of what you think you've known actually has a tremendous amount of depth to it. Coming back with fresh eyes and a better understanding of the larger picture can help you see all the beauty in it, even if this is sometimes fleeting.

I'm sorry to have vomited so much, but you know, Stockholm Syndrome is a bitch.

Claude Shannon had something to say about this as I recall.

> Do you have another suggestion? Because I don't see how code, English, or literally any other thing that can be written down is anything more meaningful than "a bunch of squiggles."

The difference is that the squiggles appear on your keyboard. If you use modern language to construct a variable name, the name can be suggestive of what it is to someone already fluent in natural language. Multi-character names also reduce collisions -- there are only so many Greek letters. And then the research paper doesn't show an inscrutable glyph as a JPEG that you can't even copy and paste into a search engine if you don't already know what it represents.

Shannon would tell you that there can only be so many simple __distinguishable__ letters. It's not just Greek. Sounds like you're mentioning the current Chinese phenomena. Where people have an easier time reading and it helps that it's text, including traditional characters. But typing can still be hard. Because that's the problem, that there's only so many easy to distinguish squiggles you can put on a keyboard. Because I'm sure while every key in distinguishable, you wouldn't want to operate a keyboard that looked like this [0] (it's actually a car decorated in keys, in case you want to see [1]).

There's a balance that that's hard to strike. The reason many languages use alphabets is not because the complexity of writing hieroglyphics, but because hieroglyphics even when written well can be difficult to distinguish and be cumbersome. There's probably no universal language (sorry Chomsky) so there's no symbols you can put on those keys that are universally recognizable. If you're actually interested in this kind of topic there's two things I suggest you look into. The first is Korean (specifically Hangul) [2] which has an interesting history relevant here and the other is long term nuclear storage [3,4]. The former was specifically designed to increase literacy and was extremely effective. The later is the difficulties of universal innate communications where we think certain things are absolutely obvious but aren't.

The other problem with languages is just usage. There's many over loaded words and it's not like we're at a shortage of combinations in the English language. Even combination of short words! The word cb doesn't exist, nor cbra! This alone should make someone suspicious that the depth of the problem is so much more than symbol complexity and combinatorics.

I'm sure Shannon would have something to say about the tyranny about encoding and complexity... Your suggesting is just the other end of the problem.

[0] https://www.ocregister.com/wp-content/uploads/migration/m1m/...

[1] https://www.ocregister.com/2012/03/30/keyboard-decorated-car...

[2] https://en.m.wikipedia.org/wiki/Hangul

[3] https://en.m.wikipedia.org/wiki/Long-term_nuclear_waste_warn...

[4] https://en.m.wikipedia.org/wiki/Long-term_nuclear_waste_warn...

All you need to do is replace the U-net with a transformer encoder (remove the embeddings, and project the image patches into vectors of size n_embd), and the diffusion process can remain the same.

[1] https://yang-song.net/blog/2021/score/

[2] https://lilianweng.github.io/posts/2021-07-11-diffusion-mode...

To gain a deeper understanding of diffusion models, I encourage everyone to read all of these blog posts and learn about the different interpretations :)

Immediately reminded of Iterative alpha-(de)Blending [1] which also sets out to set up a conceptually simpler diffusion model, and also arrives at formulating it as an approximate iterative projection process - I think this approach allows for more interesting experiments like the denoiser error analysis, though.

[1] https://arxiv.org/pdf/2305.03486.pdf

For example, what is it about image generators makes it hard for them to generate piano keyboards? It seems like some better representation of medium-distance constraints is needed to get alternating groups of two and three black notes.

I have trouble seeing reinforcement learning as convolution, for example.