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Deep Learning Is Applied Topology

theahura.substack.com

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Re: Deep Learning Is Applied Topology

#111

Post by srean »

mitthrowaway2 wrote: post #90
srean wrote: post #51

Earlier quoted context omitted.

In Boltzmann's formulation of stat-mech it comes from the assumption that when a system is in "equilibrium", then all the micro-states that are consistent with the macro-state are equally occupied. That's the basis of the theory. A prime mover is thermal agitation. It can be circular if one defines equilibrium to be that situation when all the micro-states are equally occupied. One way out is to define equilibrium in…

The Bayesian reframing of that would be that when all you have measured is the macrostate, and you have no further information by which to assign a higher probability to any compatible microstate than any other, you follow the principle of indifference and assign a uniform distribution.

Yes indeed, thanks for pointing this out. There are strong relationships between max-ent and Bayesian formulations.

For example one can use a non-uniform prior over the micro-states. If that prior happens to be in the Darmois-Koopman family that implicitly means that there are some non explicitly stated constraints that bind the micro-state statistics.

Re: Deep Learning Is Applied Topology

#112

Post by winwang »

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…

hey chris, I found your posts quite inspiring back then, with very poetic ideas. cool to see you follow up here!

Re: Deep Learning Is Applied Topology

#113

Post by HarHarVeryFunny »

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...

When you say ML, I assume you really mean LLMs?

Even with LLMs, there's no real mystery about why they work so well - they produce human-like input continuations (aka "answers") because they are trained to predict continuations of human-generated training data. Maybe we should be a bit surprised that the continuation signal is there in the first place, but given that it evidentially is, it's no mystery that LLMs are able to use it - just testimony to the power of the Transformer as a predictive architecture, and of course to gradient descent as a cold unthinking way of finding an error minimum.

Perhaps you meant how LLMs work, rather than why they work, but I'm not sure there's any real mystery there either - the transformer itself is all about key-based attention, and we now know that training a transformer seems to consistently cause it to leverage attention to learn "induction heads" (using pairs of adjacent attention heads) that are the main data finding/copying primitive they use to operate.

Of course knowing how an LLM works in broad strokes isn't the same as knowing specifically how it is working in any given case, how is it transforming a specific input layer by layer to create the given output, but that seems a bit like saying that because I can't describe - precisely - why you had pancakes for breakfast, that we don't know how the brains works.

Re: Deep Learning Is Applied Topology

#114

Post by ada1981 »

For the last few years, I've been "seeing maps" whenever I think about LLMs. It's always felt like the most natural way to understand what is going on.

It's also for this reason that I think new knowledge is discoverable from with in LLMs.

I imagine having a topographic map of some island that has only been explored partially by humans. But if I know the surrounding topography, I can make pretty accurate guesses about the areas I haven't been. And I think the same thing can be applied to certain areas of human knowledge, especially when represented as text or symbolically.

Re: Deep Learning Is Applied Topology

#115

Post by srean »

theahura wrote: post #78
ComplexSystems wrote: post #20

I really liked this article, though I don't know why the author is calling the idea of finding a separating surface between two classes of points "topology." For instance, they write "If you are trying to learn a translation task — say, English to Spanish, or Images to Text — your model will learn a topology where bread is close to pan, or where that picture of a cat is close to the word cat." This is everything that…

fair, I was using 'topology' more colloquially in that sentence. Should have said 'surface'.

Ah! That clears it up.

You then mean Deep Learning has a lot in common with differential geometry and manifolds in general. That I will definitely agree with. DG and manifolds have far richer and informative structure than topology.

Re: Deep Learning Is Applied Topology

#117

Post by srean »

whatever1 wrote: post #66
kwertzzz wrote: post #28

Earlier quoted context omitted.

Can you give an example where theories and techniques from other fields are reinvented? I would be genuinely interested for concrete examples. Such "reinventions" happen quite often in science, so to some degree this would be expected.

I mean the entire domain of systems control is being reinvented by deep RL. System identification, stability, robustness etc

Good one. Slightly different focus but they really are the same topic. Historically, Control Theory has focused on stability and smooth dynamics while RL has traditionally focused on convergence of learning algorithms in discrete spaces.

Re: Deep Learning Is Applied Topology

#118

Post by quantadev »

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.

Re: Deep Learning Is Applied Topology

#119

Post by benl »

theahura wrote: post #81
sota_pop wrote: post #4

I’ve always enjoyed this framing of the subject, the idea of mapping anything as hyperplanes existing in a solution space is one of the ideas that really blew my hair back during my academic studies. I would nitpick at your “dots in a circle example - with the stoner reference joke” I could be mistaken, but common practice isn’t to “move to a higher dimension”, but use a kernel (i.e. parameterize the points into the…

I'm pulling directly from Chris Olah's blog post with that example. But I will say that in practice, its always surprising how increasing the dimensionality of a neural network magically solves all sorts of problems. You could use a kernel if you don't have more computation available, but given more computation adding a dimension is strictly more flexible (and is capable of separating a much wider range of datasets)

Your explanation of finding a surface to separate good reasoning traces from bad reasoning traces in a high dimensional space worked as a great framing of the problem. It seems though that the surface will be fractal - the distance between a good trace and a bad trace could be arbitrarily small. If so then the work required to find and compute better and better surfaces will grow arbitrarily large. I wonder if there is a rigorous way to determine if the surface is fractal or not.

Re: Deep Learning Is Applied Topology

#120

Post by riemannzeta »

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 is near a critical point, the lattice topology doesn't even matter.

It's only an analogy, but it does suggest at least that the interesting details of the dynamics aren't embedded in the topology of the system. It's more complicated than that.

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