[adamnemecek]

I have recently written a paper on understanding machine learning via the lens of Hopf algebra https://arxiv.org/abs/2302.01834.

Hopf algebras (which are really just tensors with recurrence relations built in) subsume convnets, transformers and diffusion model and also provide a theoretically better autodiff that operates within single layers as opposed to across entire graphs.

Furthermore, there is a correspondence between Hopf algebra and cyclical linear logic and Hopf algebras are related to zonotopes, which are polyhedra that have been used in verified numerical computation. I'm strongly convinced the LL connection can provide proofs over zonotopes which paves the way towards interpretable AI and will be central for XAI.

I know this sounds too good to be true but Persi Diaconis has also written a paper that shows how useful Hopf algebras are in the context of Markov chains https://arxiv.org/abs/1206.3620

I'm working on a next gen Hopf algebra based machine learning framework.

Join my discord if you want to discuss this further https://discord.cofunctional.ai.

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My account is currently rate limited so I will use this comment to respond to comments below.

red_trumped: What about Hopf algebras do I not understand?

gaze: Haha, it's been a while since I have commented about QC. What do I not understand about it? And what comment are you referring to?

[red_trumpet]

Your paper didn't pass my smell test at all, tbh. For example the formula you write about "product" and "coproduct" in section 3 is literally identical (as "=" is symmetric). In section 4.2 you write "the product is the standard tensor product" with a formula that doesn't at all involve the map m: A \otimes A \to A. The formula you write is the induced product on A \otimes A, assuming that you already have a product on A. The formula for "coproduct" is just an example[1] of a coproduct, not every coproduct has to look that way.

[1] https://en.wikipedia.org/wiki/Coalgebra#Examples

[keithalewis]

Same here. Sloppily written with very little content. If the author can't take the time to proofread his own paper why should anyone else waste their time?

[adamnemecek]

Ok read the Persi Diaconis one https://arxiv.org/abs/1206.3620.

You are right, it can be more polished but also it covers quite a lot and it was surprisingly hard to fit it all in.

The main response I have been getting though is "I won't believe this until I see the implementation" so I have been concentrating on that.

[jurynulifcation]

adamnemecek has posted too many comments and is in cooldown phase, but he's asked me to post this comment: "It's the programmers equal sign. I think that the surrounding text provides a decent explanation what the deal is.

You are right, there's a missing sentence fragment, "standard tensor product that satisfies the property...".

Read the Diaconis paper. "

--- This isn't a sock puppet and I hope this isn't against site rules. I just wanted to try and help facilitate good discussion. I think trumpet brought up some interesting criticisms and felt Adam had a legitimate interest in responding ASAP.

[LudwigNagasena]

> It's the programmers equal sign.

That doesn’t seem to make any sense.

[adamnemecek]

I'm saying there's a difference between equal sign that defines something and equal sign that declares a relation.

[red_trumpet]

Sure. In mathematics this is why we often use ":=" for definitions, or we indicate in the surrounding text that the next equation is a definition. That would be helpful.

But even then, you cannot define "C" as "C \otimes C", because the right hand side only makes sense if "C" is already defined. And in math you cannot define something twice. As soon as you defined something, it stays the same in the given context.

[adamnemecek]

It's a type signature, not a formula. Cs are not values.

Think of it as a signature in say Rust

fn coproduct(in: C) -> (C, C)

Also you were talking about my knowledge of Hopf algebras, this is not knowledge of Hopf algebra but quibbling about things that are pretty clear from context.

[scatters]

It seems fine to me, although maybe it's backwards. Add a prime to the RHS, perhaps.

[gexaha]

could you advertise your research a bit less often, please? i see your post like literally almost every other day here

[hgsgm]

Or at least explain it in more accessible way. Every time Adam posts about the paper, it gets confused comments and no engagement on the content, because it's pretty deep graduate level pure math, which is occasionally seen but rare on HN.

[red_trumpet]

As a maths PhD student that has seen Hopf algebras before (though I'm no expert, and the context was different), I'm not convinced Adam understands things about Hopf algebras.

[gaze]

He doesn’t understand quantum computing either. I wish he would stop writing authoritatively about things he doesn’t understand.

[bawolff]

I was kind of curious what all the controversy is about, but looking back at that users comments, it doesn't seem like he has even commented once in the last six months about QC, unless i missed it (?)

[adamnemecek]

At some point I said that analog QC will be the future rather than what's considered QC today. I still stand by those comments.

[adamnemecek]

What about QC do I not understand? It's been a while since I have made any comments about QC so I'm glad that those comments made an impression.

[gaze]

I don’t think they made the kind of impression you wanted

[adamnemecek]

Go on what do I not understand about Hopf algebras?

[adamnemecek]

The discord channel is popping off. Come join, https://discord.cofunctional.ai I'll explain it

[nicwilson]

Intersting papers.

https://arxiv.org/abs/2302.01834 appears to have a typo in section 4.5

S(hg) = S(g)S(g)

looks like it should be S(hg) = S(h)S(g) or S(hg) = S(g)S(h)

[adamnemecek]

Right thanks.

[ianandrich]

I just read your Coinductive guide to inductive transformer heads paper.

My mind is blown.

Is the Hopf Algebra based ML framework you are working on on your github? I took a glance, but you have 1500 repositories and it wasn't on the first few of them.

[adamnemecek]

It's in very early stages and it's not there yet no. Join the discord https://discord.cofunctional.ai or my twitter https://twitter.com/adamnemecek1 if you want to follow progress. It might take some time.