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Deep Learning Is Applied Topology

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Re: Deep Learning Is Applied Topology

#161

Post by mjburgess »

jvanderbot wrote: post #139
mjburgess wrote: post #135

Earlier quoted context omitted.

People are only mushy in their verbalised reasoning, because its the nature of such reasoning to handle hard cases. Animal cognition, at its basic levels, is incredibly refined and makes necessary use of logic, flawlessly, frequently. This naive cynicism about our mental capacities is a product of this credulity about statistical AI. If one beings with an earnest study of animal intelligence, in order to describe it,…

Well, we disagree fundamentally. And, I applaud the heavy handed use of condescension. Logical propositions ("2+2=4 regardless of my certainty about it") seem a long way from necessary or sufficient to survival for animals. A fuzzy heatmap of "where is prey going" or "How many prey over there" is much closer to necessary and sufficient. The fact that measurements or senses can update those estimates is a long way fro…

You can enumerate, all you wish, all the fuzzy judgements we need to make. This confirms a capacity for uncertain reasoning. It says nothing about the trivial and innumerable ways concepts compose both in content (imagine that A and-also B) and in logical relation (eg., imagine that not A and B).

The point of my "condescension" was to point out that people of your position are arguing from ignorance, with confirmation bias -- ie., having no study of animal intelligence, and only ever repeating what they know about their own study of irrelevant systems.

Your reply evidences this exactly. Zero engagement with any facts on the ground about actual animal intelligence. Are you really actually trying to account for animal intelligence, or as I have claimed twice now, are you only really wishing to maintain your ignorance of it, dismiss any analysis of it, and instead "confirm" that whatever you are aware of "must, presumably, apply".

Imagine being faced with such bad faith over and over and over again. It is like arguing with people who insist the world is flat, and when challenged, point to euclidean geometry and the flatness of the pavement under their feet. If I begin by anticipating such behaviour, you can see why.

Re: Deep Learning Is Applied Topology

#162

Post by mjburgess »

naasking wrote: post #158
mjburgess wrote: post #129

Earlier quoted context omitted.

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.

> It's also vastly energetically cheaper just to have (algorithmic) negation. Even if true, that's an argument that it's cheaper to have something, not that it's cheaper to develop it through natural selection. Training time and energy for LLMs shows how energy intensive training to get to the point of grokking/circuit generalization.

It is a matter of empirical fact that we can reason with logical relationships. Thus taking an LLM and it's training as a model of conginition is empriically false.

It should be obviously doubly so, since as a model -- as you point out -- it makes trivial aspects of our cognition impossibly expensive to acqurie.

Re: Deep Learning Is Applied Topology

#163

Post by riemannzeta »

colah3 wrote: post #128
riemannzeta wrote: post #120

Earlier quoted context omitted.

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/

Thanks for this additional link, which really underscores for me at least how you're right about patterns in circuits being a better abstraction layer for capturing interesting patterns than topological manifolds.

I wasn't familiar with the term "equivariance" but I "woke up" to this sort of approach to understanding deep neural networks when I read this paper, which shows how restricted boltzman machines have an exact mapping to the renormalization group approach used to study phase transitions in condensed matter and high energy physics:

https://arxiv.org/abs/1410.3831

At high enough energy, everything is symmetric. As energy begins to drain from the system, eventually every symmetry is broken. All fine structure emerges from the breaking of some symmetries.

I'd love to get more in the weeds on this work. I'm in my own local equilibrium of sorts doing much more mundane stuff.

Re: Deep Learning Is Applied Topology

#164

Post by somewhereoutth »

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…

Regardless of the mechanism, the foundational 'conceit' of LLMs is that by dumping enough syntax (and only syntax) into a sufficiently complex system, the semantics can be induced to emerge.

Quite a stretch, in my opinion (cf. Plato's Cave).

Re: Deep Learning Is Applied Topology

#165

Post by j2kun »

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…

This has mirrored my experience attempting to "apply" topology in real world circumstances, off and on since I first studied topology in 2011.

I even hesitate now at the common refrain "real world data approximates a smooth, low dimensional manifold." I want to spend some time really investigating to what extent this claim actually holds for real world data, and to what extent it is distorted by the dimensionality reduction method we apply to natural data sets in order to promote efficiency. But alas, who has the time?

Re: Deep Learning Is Applied Topology

#167

Post by 3abiton »

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…

The linear representation hypothesis is rather quite intreguing, I am curious what was the intuition behind it.

Re: Deep Learning Is Applied Topology

#168

Post by steppi »

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…

If I had to give a loose definition of topology, I would say that it is actually about studying spaces which have some notion of what is close and far, even if no metric exists. The core idea of neighborhoods in point set topology captures the idea of points being nearby another point, and allows defining things like continuity and sequence convergence which require a notion of closeness. From Wikipedia [0] for example

The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.

Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

That's not to say that topology is necessarily the best lens for understanding neural networks, and the article's author has shown up in the comments to state he's moved on in his thinking. I'm just trying to clear up a misconception.

[0] https://en.wikipedia.org/wiki/General_topology

Re: Deep Learning Is Applied Topology

#169

Post by colah3 »

3abiton wrote: post #167
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…

The linear representation hypothesis is rather quite intreguing, I am curious what was the intuition behind it.

See https://transformer-circuits.pub/2022/toy_model/index.html#m...

If you're new to this, I'd mostly just look at all the empirical examples.

The slightly harder thing is to consider the fact that neural networks are made of linear functions with non-linearities between them, and to try to think about when linear directions will be computationally natural as a result.

Re: Deep Learning Is Applied Topology

#170

Post by strimp099 »

One of the most successful quant traders in history, Jim Simons studied topology in the 60s. It’s rumored he used deep neural networks in his trading before they were cool. This post really brought the two together for me in a way I didn’t understand before.
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