Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Deep Learning Is Applied Topology

theahura.substack.com

Post reply on HN
Print view View original

121–130 of 200 posts

Re: Deep Learning Is Applied Topology

#121

Post by srean »

quantadev wrote: post #118
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…

The word "topology" has a legitimate dictionary definition, that has none of the requirements that you're asserting. I think what you're missing is that it has two definitions.

In blog posts about specialised and technical topics it is expected that in-domain technical keywords that have long established definitions and meanings be used in the same technical sense. Otherwise it can become quite confusing. Gravity means gravity when we are talking Newtonian mechanics. Similarly, in math and ML 'topology' has a specific meaning.

Re: Deep Learning Is Applied Topology

#122

Post by theahura »

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…

Thanks for the follow up. I've been following your circuits thread for several years now. I find the linear representation hypothesis very compelling, and I have a draft of a review for Toy Models of Superposition sitting in my notes. Circuits I find less compelling, since the analysis there feels very tied to the transformer architecture in specific, but what do I know.

Re linear representation hypothesis, surely it depends on the architecture? GANs, VAEs, CLIP, etc. seem to explicitly model manifolds. And even simple models will, due to optimization pressure, collapse similar-enough features into the same linear direction. I suppose it's hard to reconcile the manifold hypothesis with the empirical evidence that simple models will place similar-ish features in orthogonal directions, but surely that has more to do with the loss that is being optimized? In Toy Models of Superposition, you're using a MSE which effectively makes the model learn an autoencoder regression / compression task. Makes sense then that the interference patterns between co-occurring features would matter. But in a different setting, say a contrastive loss objective, I suspect you wouldn't see that same interference minimization behavior.

Re: Deep Learning Is Applied Topology

#123

Post by theahura »

kentuckyrobby wrote: post #100
theahura wrote: post #75

Earlier quoted context omitted.

> Topology is useful especially when you cannot trust lengths, distances angles and arbitrary deformations But...you can't. The input data lives on a manifold that you cannot 'trust'. It doesn't mean anything apriori that an image of a coca-cola can and an image of a stopsign live close to each other in pixel space. The neural network applies all of those violent transformations you are talking about

I don't think that was their point, I think their point was that neural networks 'create' their optimization space by using lengths, distances, and angles. You can't reframe it from a topological standpoint, otherwise optimization spaces of some similar neural networks on similar problems would topologically comparable, which is not true.

Well, sorta. There is some evidence to suggest that neural networks learn 'universal' features (cf Anthropic's circuits thread). But I'll openly admit to being out of my depth here, and maybe I just don't understand OPs point

Re: Deep Learning Is Applied Topology

#124

Post by naasking »

mjburgess wrote: post #35
soulofmischief wrote: post #26

Thanks for sharing. I also tend to view learning in terms of manifolds. It's a powerful representation. > I'm personally pretty convinced that, in a high enough dimensional space, this is indistinguishable from reasoning I actually have journaled extensively about this and even written some on Hacker News about it with respect to what I've been calling probabilistic reasoning manifolds: > This manifold is constructed…

Are you talking about reasoning in general, reasoning qua that mental process which operates on (representations of) propositions? In which case, I cannot understand " true reasoning is expressed in terms of probabilities, not axioms " One of the features of reasoning is that it does not operate in this way. It's highly implausible animals would have been endowed with no ability to operate non-probabilistically on pr…

> It's highly implausible animals would have been endowed with no ability to operate non-probabilistically on propositions represented by them, since this is essential for correct reasoning

Why would animals need to evolve 100% correct reasoning if probabilistically correct reasoning suffices? If probabilistic reasoning is cheaper in terms of energy then correct reasoning is a disadvantage.

Re: Deep Learning Is Applied Topology

#125

Post by iNic »

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…

My guess is that the linear representation hypothesis is only approximately right in the sense that my expectation is that it is more like a Lie Group. Locally flat, but the concept breaks at some point. Note that I am a mathematician who knows very little about machine learning apart from taking a few classes at uni

Re: Deep Learning Is Applied Topology

#127

Post by colah3 »

theahura wrote: post #122
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…

Thanks for the follow up. I've been following your circuits thread for several years now. I find the linear representation hypothesis very compelling, and I have a draft of a review for Toy Models of Superposition sitting in my notes. Circuits I find less compelling, since the analysis there feels very tied to the transformer architecture in specific, but what do I know. Re linear representation hypothesis, surely it…

> Circuits I find less compelling, since the analysis there feels very tied to the transformer architecture in specific, but what do I know.

I don't think circuits is specific to transformers? Our work in the Transformer Circuits thread often is, but the original circuits work was done on convolutional vision models (https://distill.pub/2020/circuits/ )

> Re linear representation hypothesis, surely it depends on the architecture? GANs, VAEs, CLIP, etc. seem to explicitly model manifolds

(1) There are actually quite a few examples of seemingly linear representations in GANs, VAEs, etc (see discussion in Toy Models for examples).

(2) Linear representations aren't necessarily in tension with the manifold hypothesis.

(3) GANs/VAEs/etc modeling things as a latent gaussian space is actually way more natural if you allow superposition (which requires linear representations) since central limit theorem allows superposition to produce Gaussian-like distributions.

Re: Deep Learning Is Applied Topology

#128

Post by colah3 »

riemannzeta wrote: post #120
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…

I think it's interesting that in physics, different global symmetries (topological manifolds) can satisfy the same metric structure (local geometry). For example, the same metric tensor solution to Einstein's field equation can exist on topologically distinct manifolds. Conversely, looking at solutions to the Ising Model, we can say that the same lattice topology can have many different solutions, and when the system…

If you like symmetry, you might enjoy how symmetry falls out of circuit analysis of conv nets here:

https://distill.pub/2020/circuits/equivariance/

Re: Deep Learning Is Applied Topology

#129

Post by mjburgess »

naasking wrote: post #124
mjburgess wrote: post #35

Earlier quoted context omitted.

Are you talking about reasoning in general, reasoning qua that mental process which operates on (representations of) propositions? In which case, I cannot understand " true reasoning is expressed in terms of probabilities, not axioms " One of the features of reasoning is that it does not operate in this way. It's highly implausible animals would have been endowed with no ability to operate non-probabilistically on pr…

> It's highly implausible animals would have been endowed with no ability to operate non-probabilistically on propositions represented by them, since this is essential for correct reasoning Why would animals need to evolve 100% correct reasoning if probabilistically correct reasoning suffices? If probabilistic reasoning is cheaper in terms of energy then correct reasoning is a disadvantage.

It doesnt suffice. It's also vastly energetically cheaper just to have (algorithmic) negation. Compressing (A, not A) into a probability function is extremely incomprehensibly expensive.

Re: Deep Learning Is Applied Topology

#130

Post by cvoss »

quantadev wrote: post #118
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…

The word "topology" has a legitimate dictionary definition, that has none of the requirements that you're asserting. I think what you're missing is that it has two definitions.

The phrase "applied X" invokes the technical, scientific, or academic meaning of X. So for example, "applied chemistry" does not refer to one's experience on a dating app.
Post reply on HN