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Deep Learning Is Applied Topology

theahura.substack.com

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Re: Deep Learning Is Applied Topology

#101

Post by nis0s »

> One way to think about neural networks, especially really large neural networks, is that they are topology generators. That is, they will take in a set of data and figure out a topology where the data has certain properties. Those properties are in turn defined by the loss function.

Latent spaces may or may not have useful topology, so this idea is inherently wrong, and builds the wrong type of intuition. Different neural nets will result in different feature space understanding of the same data, so I think it's incorrect to believe you're determining intrinsic geometric properties from a given neural net. I don't think people should throw around words carelessly because all that does is increase misunderstanding of concepts.

In general, manifolds can help discern useful characteristics about the feature space, and may have useful topological structures, but trying to impose an idea of "topology" on this is a stretch. Moreover, the kind of basics examples used in this blog post don't help prove the author's point. Maybe I am misunderstanding this author's description of what they mean, but this idea of manifold learning is nothing new.

Re: Deep Learning Is Applied Topology

#102

Post by Graviscalar »

theahura wrote: post #96
Graviscalar wrote: post #71

I was one of the people that was super excited after reading the Chris Olah blogpost from 2014, and over the past decade I've seen the insight go exactly nowhere. It's neat but it hasn't driven any interesting results, though Ayasdi did some interesting stuff with TDA and Gunnar Carlson has been playing around with neural nets recently.

I think it's incorrect that the insight has gone nowhere. See, for eg, contrastive loss / clip, or vqgan image generation. Arguably also diffusion models. More generally, in my experience as an AI researcher, understandings of the geometry of data leads directly to changes in model architecture. Though people disparage that as "trial and error" it is far more directed than people on the outside give credit for.

The geometric intuition is solid, but actually applying topology has been less fruitful in spite of a lot of people trying their best, as Chris Olah himself has said elsewhere in this thread.

Re: Deep Learning Is Applied Topology

#104

Post by deepburner »

This whole article is just a nothingburger. Saying something is applied topology is only one step more advanced than saying something is maths - duh. These mathematical abstractions are incredibly general and and you can pretty much draw up anything in terms of anything, the challenging part is being able to turn around and use the model/abstraction to say things about the thing you're abstracting. I don't think scholars have been very successful in that regard, less so this article.

Yeah deep learning is applied topology, it's also applied geometry, and probably applied algebra and I wouldn't be surprised if it was also applied number theory.

Re: Deep Learning Is Applied Topology

#105

Post by wenc »

qbit42 wrote: post #74
wenc wrote: post #54

Earlier quoted context omitted.

Many types of data don’t. Disconnected spaces like integer spaces don’t sit on a manifold (they are lattices). Spiky noisy fragmented data don’t sit on a (smooth) manifold. In fact not all ML models treat data as manifolds. Nearest neighbors, decision trees don’t require the manifold assumption and actually work better without it.

Any reasonable statistical explanation of deep learning requires there to be some sort of low dimensional latent structure in the data. Otherwise, we would not have enough training data to learn good models, given how high the ambient dimensions are for most problems.

Deep learning specifically yes. It needs a manifold assumption. But not data in general which was what I was responding to.

Re: Deep Learning Is Applied Topology

#106

Post by wenc »

theahura wrote: post #92
wenc wrote: post #54

Earlier quoted context omitted.

Many types of data don’t. Disconnected spaces like integer spaces don’t sit on a manifold (they are lattices). Spiky noisy fragmented data don’t sit on a (smooth) manifold. In fact not all ML models treat data as manifolds. Nearest neighbors, decision trees don’t require the manifold assumption and actually work better without it.

It turns out a lot of disconnected spaces can be approximated by smooth ones that have really sharp boundaries, which more or less seems to be how neural networks will approximate something like discrete tokens

Can be approximated yes. Approximated well? No, but you can get away with it sometimes with saturation functions like softmax. But badly. It’s like trying to solve an integer program as a linear program. You end up with a relaxation that is not integral and not the real answer.

An integer lattice can only be a manifold in a trivial sense (dimension 0). But not for any positive dimensions.

Re: Deep Learning Is Applied Topology

#107

Post by throwawaymaths »

srean wrote: post #29

The title, as it stands, is trite and wrong. More about that a little later. The article on the other hand is a pleasant read. Topology is whatever little structure that remains in geometry after you throwaway distances, angles, orientations and all sorts of non tearing stretchings. It's that bare minimum that still remains valid after such violent deformations. While notion of topology is definitely useful in machin…

once you get into the nitty gritty, a lot of things that wouldn't matter if it were pure topology, do, like number of layers all the way to quantization/fp resolution

Re: Deep Learning Is Applied Topology

#108

Post by montebicyclelo »

colah3 wrote: post #98

Since this post is based on my 2014 blog post ( https://colah.github.io/posts/2014-03-NN-Manifolds-Topology/ ), I thought I might comment. I tried really hard to use topology as a way to understand neural networks, for example in these follow ups: - https://colah.github.io/posts/2014-10-Visualizing-MNIST/ - https://colah.github.io/posts/2015-01-Visualizing-Representa... There are places I've found the topological per…

Related to ways of understanding neural networks, I've seen these views expressed a lot, which to me seem like misconceptions:

- LLMs are basically just slightly better `n-gram` models

- The idea of "just" predicting the next token, as if next-token-prediction implies a model must be dumb

(I wonder if this [1] popular response to Karpathy's RNN [2] post is partly to blame for people equating language neural nets with n-gram models. The stochastic parrot paper [3] also somewhat equates LLMs and n-gram models, e.g. "although she primarily had n-gram models in mind, the conclusions remain apt and relevant". I guess there was a time where they were more equivalent, before the nets got really really good)

[1] https://nbviewer.org/gist/yoavg/d76121dfde2618422139

[2] https://karpathy.github.io/2015/05/21/rnn-effectiveness/

[3] https://dl.acm.org/doi/pdf/10.1145/3442188.3445922

Re: Deep Learning Is Applied Topology

#109

Post by nomel »

Regic wrote: post #60
umutisik wrote: post #3

Data doesn't actually live on a manifold. It's an approximation used for thinking about data. Near total majority, if not 100%, of the useful things done in deep learning have come from not thinking about topology in any way. Deep learning is not applied anything, it's an empirical field advanced mostly by trial and error and, sure, a few intuitions coming from theory (that was not topology).

I feel like the fact that ML has no good explanation why it works this well gives a lot of people room to invent their head-canon, usually from their field of expertise. I've seen this from exceptionally intelligent individuals too. If you only have a hammer...

I think it would be more unusual, and concerning, if an intelligent individual didn't attempt to apply their expertise for a head-canon of something unknown.

Coming up with an idea for how something works, by applying your expertise, is the fundamental foundation of intelligence, learning, and was behind every single advancement of human understanding.

People thinking is always a good thing. Thinking about the unknown is better. Thinking with others is best, and sharing those thoughts isn't somehow bad, even if they're not complete.

Re: Deep Learning Is Applied Topology

#110

Post by fusionadvocate »

theahura wrote: post #80
terabytest wrote: post #45

I'm confused by the author's diagram claiming that AGI/ASI are points on the same manifold as next token prediction, chat models, and CoT models. While the latter three are provably part of the same manifold, what justifies placing AGI/ASI there too? What if the models capable of CoT aren't and will never be, regardless of topological manipulation, capable of processes that could be considered AGI? For example, human…

Possible. AGI/ASI are poorly defined. I tend to think we're already at AGI, obviously many disagree. > For example, human intelligence (the closest thing we know to AGI) requires extremely complex sensory and internal feedback loops and continuous processing unlike autoregressive models' discrete processing. I've done a fair bit of connectomics research and I think that this framing elides the ways in which neural ne…

I believe we already have the technology required for AGI. It perhaps is analogous to a lunar manned station or a 2 mile tall skyscrapper. We have the technology required to build it, but we don't for various reasons.
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