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Deep Learning Is Applied Topology

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Re: Deep Learning Is Applied Topology

#141

Post by godelski »

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…

Loved these posts and they inspired a lot of my research and directions during my PhDs.

For anyone interested in these may I also suggest learning about normalizing flows? (They are the broader class to flow matching) They are learnable networks that learn coordinate changes. So the connection to geometry/topology is much more obvious. Of course the down side of flows is you're stuck with a constant dimension (well... sorta) but I still think they can help you understand a lot more of what's going on because you are working in a more interpretable environment

Re: Deep Learning Is Applied Topology

#142

Post by colah3 »

montebicyclelo wrote: post #108
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…

I guess I'll plug my hobby horse:

The whole discourse of "stochastic parrots" and "do models understand" and so on is deeply unhealthy because it should be scientific questions about mechanism, and people don't have a vocabulary for discussing the range of mechanisms which might exist inside a neural network. So instead we have lots of arguments where people project meaning onto very fuzzy ideas and the argument doesn't ground out to scientific, empirical claims.

Our recent paper reverse engineers the computation neural networks use to answer in a number of interesting cases (https://transformer-circuits.pub/2025/attribution-graphs/bio... ). We find computation that one might informally describe as "multi-step inference", "planning", and so on. I think it's maybe clarifying for this, because it grounds out to very specific empirical claims about mechanism (which we test by intervention experiments).

Of course, one can disagree with the informal language we use. I'm happy for people to use whatever language they want! I think in an ideal world, we'd move more towards talking about concrete mechanism, and we need to develop ways to talk about these informally.

There was previous discussion of our paper here: https://news.ycombinator.com/item?id=43505748

Re: Deep Learning Is Applied Topology

#143

Post by mirekrusin »

ComplexSystems wrote: post #24
mirekrusin wrote: post #11

Just because manifold looks a bit like burrito if you squint doesn't mean it is a burrito.

What if you don't have to squint very much?

Same amount of squinting needed as for claim that deep learning is just a bunch of matrices. Or 0s and 1s. Cool. Now what?

Re: Deep Learning Is Applied Topology

#144

Post by quantadev »

srean wrote: post #138
quantadev wrote: post #131

Earlier quoted context omitted.

The word "topology" is quite commonly used in all kinds of books, papers, and technical materials any time they're discussing geometric characteristics of surfaces. The term is probably used 1000000 times more commonly in this more generic way than it's ever used in the strict pedantic way you're asserting that it must.

Surfaces certainly have a topology (potentially more than one), surfaces are examples of one kind of a topological space, in fact the next interesting one after a curve. So I will not be surprised at all with co-occurrences of 'surface' and 'topology'. But surfaces and topologies mean different things. Dogs have fur. Dogs are an example of a furry animal. But dogs and furs are not the same thing although they may app…

The most common uses of "topology", whenever used to convey a geometry-related idea, is in the more general sense meaning "surfaces". Only one out of a million times is anyone ever referring to the specific mathematical field of the same name to which you refer.

Re: Deep Learning Is Applied Topology

#145

Post by garganzol »

I want to share one more related observation: by definition, topology math refers to geometrical objects and transformations. But there exists another, more computer-esque definition of topology that defines relations between abstract objects.

For example, let's take a look at graph data structure. A graph has a set of stored objects (vertices) and a set of stored relations between the vertices (edges). In this way, graph defines a topology in discrete form.

Let's take a look at network data structure which is closely related to the graph. It is very much the same idea, but it additionally has a value stored in every edge. A network has a set of objects (vertices) and a set of relations between the objects (edges), while edges also hold edge values. So it is also a form of topology because the network defines the relations between the abstract objects.

In this light, you can view a graph as a neural network with {0, 1} weights. The graph edge is either present or absent, hence {0, 1} values only. The network structure, however, can hold any assigned value in every edge, so every connection between objects (neurons) can be characterized not only by its presence, but also by edge-assigned values (weights). Now we get the full model of a neural network. And yes, it is built upon topology in its discrete form.

Re: Deep Learning Is Applied Topology

#146

Post by [deleted] »

quantadev wrote: post #133
cvoss wrote: post #130

Earlier quoted context omitted.

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.

The word "topology" is _much_ more commonly used as a general synonym for "surfaces" than in any other way.

[deleted]

Re: Deep Learning Is Applied Topology

#147

Post by srean »

quantadev wrote: post #144
srean wrote: post #138

Earlier quoted context omitted.

Surfaces certainly have a topology (potentially more than one), surfaces are examples of one kind of a topological space, in fact the next interesting one after a curve. So I will not be surprised at all with co-occurrences of 'surface' and 'topology'. But surfaces and topologies mean different things. Dogs have fur. Dogs are an example of a furry animal. But dogs and furs are not the same thing although they may app…

The most common uses of "topology", whenever used to convey a geometry-related idea, is in the more general sense meaning "surfaces". Only one out of a million times is anyone ever referring to the specific mathematical field of the same name to which you refer.

LOL the very first line of your own link

(that now you seem to have deleted after my comment https://en.m.wikipedia.org/wiki/Topology_(disambiguation) )

"Topology is a branch of mathematics concerned with geometric properties preserved under continuous deformation (stretching without tearing or gluing)"

That is indeed the established meaning of topology, more so in mathematics and the blog post was on applied mathematics. That it may mean something else in other contexts is irrelevant.

I rest my case.

> The most common uses of "topology", whenever used to convey a geometry-related idea, is in the more general sense meaning "surfaces"

Erm, citation please.

I included a search on Amazon on topology https://www.amazon.com/s?k=Topology&sprefix=topology+%2Caps%... (without even adding the keyword maths. None of the results seem to be about surfaces. Shouldn't there have been a few ? Wouldn't Amazon search results reflect the general sense meaning ?).

If it were true, wouldn't the Wikipedia pages have talked about that general sense meaning first ?

Alternatively, I would say, take a breath. Is this hill really the one worth dying on ? There are better ones. Have a good day and if work permits, get yourself a juicy topology book, it can be interesting, if presented well.

Re: Deep Learning Is Applied Topology

#148

Post by agentcoops »

montebicyclelo wrote: post #108
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…

1000%. It's really hard to express this to non-engineers who never wasted years of their life trying to work with n-grams and NLTK (even topic models) to make sense of textual data... Projects I dreamed of circa 2012 are now completely trivial. If you do have that comparison ready-at-hand, the problem of understanding what this mind-blowing leap means, to which end I find writing like the OP helpful, is so fascinating and something completely different than complaining that it's a "black box."

I've expressed this on here before, but it feels like the everyday reception of LLMs has been so damaged by the general public having just gotten a basic grasp on the existence of machine learning.

Re: Deep Learning Is Applied Topology

#149

Post by mdp2021 »

colah3 wrote: post #142
montebicyclelo wrote: post #108

Earlier quoted context omitted.

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…

I guess I'll plug my hobby horse: The whole discourse of "stochastic parrots" and "do models understand" and so on is deeply unhealthy because it should be scientific questions about mechanism, and people don't have a vocabulary for discussing the range of mechanisms which might exist inside a neural network. So instead we have lots of arguments where people project meaning onto very fuzzy ideas and the argument does…

Absolutely, the first task should be to understand how and why black boxes with emergent properties actually work, in order to further knowledge - but importantly, in order to improve them and build on the acquired knowledge to surpass them. That implies, curbing «parrot[ing]» and inadequate «understand[ing]».

I.e. those higher concepts are kept in mind as a goal. It is healthy: it keeps the aim alive.

Re: Deep Learning Is Applied Topology

#150

Post by theahura »

colah3 wrote: post #140
theahura wrote: post #134

Earlier quoted context omitted.

> the original circuits work was done on convolutional vision models O neat, I haven't read that far back. Will add it to the reading list. To flesh this out a bit, part of why I find circuits less compelling is because it seems intuitive to me that neural networks more or less smoothly blend 'process' and 'state'. As an intuition pump, a vector x matrix matmul in an MLP can be viewed as changing the basis of an inpu…

> True! I suppose I was thinking about a 'strong' form of linear representations, which is something like: features are represented by linear combinations of neurons that display the same repulsion-geometries as observed in Toy Models, but that's not what you're saying / that's me jumping a step too far. Note this happens in "uniform superposition". In reality, we're almost certainly in very non-uniform superposition…

> One key term to look for is "feature manifolds" or "multi-diemsnional features"

I should probably read the updates more. Not enough time in the day. But yea the way you're describing feature manifolds and multidimensional features, especially the importance of linearity-in-properties and not necessarily linearity-in-dimensions, makes a lot of sense and is basically how I default think about these things.

> but doesn't quite capture the technical nuance. So it's hard to know the full scope of what you intend.

Fair, I'm only passingly familiar with compressed sensing so I'm not sure I could offer a more technical definition without, like, a much longer conversation! But it's good to know in the future that in a technical sense linear representations and superposition are dependent.

> all features are linear, rather than just most or some

Potentially a tangent, but compared to what? I suppose the natural answer is "non linear features" but has there been anything to suggest that neural networks represent concepts in this way? I'd be rather surprised if they did within a single layer. (Across layers, sure, but that actually starts to pull me more towards circuits)

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