Deep Learning Is Applied Topology

https://theahura.substack.com/p/deep-learning-is-applied-topology

by theahura • 1 year ago

508 182 1 year ago

theahura.substack.com

esafak3• 1 year ago

If it was topology we wouldn't bother to warp the manifold so we can do similarity search. No, it's geometry, with a metric. Just as in real life, we want to be able to compare things.

Topological transformation of the manifold happens during training too. That makes me wonder: how does the topology evolve during training? I imagine it violently changing at first before stabilizing, followed by geometric refinement. Here are some relevant papers:

* Topology and geometry of data manifold in deep learning (https://arxiv.org/abs/2204.08624)

* Topology of Deep Neural Networks (https://jmlr.org/papers/v21/20-345.html)

* Persistent Topological Features in Large Language Models (https://arxiv.org/abs/2410.11042)

* Deep learning as Ricci flow (https://www.nature.com/articles/s41598-024-74045-9)

profchemai2• 1 year ago

Agree, if anything it's Applied Linear Algebra...but that sounds less exotic.

lostmsu• 1 year ago

Well, we know it is non-linear. More like differential equations.

theahura16• 1 year ago

Topological transformation of the manifold happens during training too. That makes me wonder: how does the topology evolve during training?

If you've ever played with GANs or VAEs, you can actually answer this question! And the answer is more or less 'yes'. You can look at GANs at various checkpoints during training and see how different points in the high dimensional space move around (using tools like UMAP / TSNE).

I imagine it violently changing at first before stabilizing, followed by geometric refinement

Also correct, though the violent changing at the beginning is also influenced the learning rate and the choice of optimizer.

esafak3• 1 year ago

And crucially, the initialization algorithm.

umutisik2• 1 year ago

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).

thuuuomas• 1 year ago

I cannot understand this prideful resentment of theory common among self-described practitioners.

Even if existing theory is inadequate, would an operating theory not be beneficial?

Or is the mystique combined with guess&check drudgery job security?

hiddencost• 1 year ago

Maybe a little less with the ad hominems? The OP is providing an accurate description of an extremely immature field.

jebarker2• 1 year ago

There are strong incentives to leave theory as technical debt and keep charging forward. I don't think it's resentment of theory, everyone would love a theory if one were available but very few are willing to forgoe the near term rewards to pursue theory. Also it's really hard.

canjobear• 1 year ago

If there were theory that led to directly useful results (like, telling you the right hyperparameters to use for your data in a simple way, or giving you a new kind of regularization that you can drop in to dramatically improve learning) then deep learning practitioners would love it. As it currently stands, such theories don't really exist.

lumost• 1 year ago

There are many reasons to believe a theory may not be forthcoming, or that if it is available may not be useful.

For instance, we do not have consensus on what a theory should accomplish - should it provide convergence bounds/capability bounds? Should it predict optimal parameter counts/shapes? Should it allow more efficient calculation of optimal weights? Does it need to do these tasks in linear time?

Even materials science in metals is still cycling through theoretical models after thousands of years of making steel and other alloys.

danielmarkbruce• 1 year ago

Who is proud? What you are seeing in some cases is eye rolling. And it's fair eye rolling.

There is an enormous amount of theory used in the various parts of building models, there just isn't an overarching theory at the very most convenient level of abstraction.

It almost has to be this way. If there was some neat theory, people would use it and build even more complex things on top of it in an experimental way and then so on.

motoboi2• 1 year ago

Your comment sits in the nice gradient between not seeing at all the obvious relationships between deep learning and topology and thinking that deep learning is applied topology.

See? Everything lives in the manifold.

Now for a great visualization about the Manifold Hypothesis I cannot recommend more this video: https://www.youtube.com/watch?v=pdNYw6qwuNc

That helps to visualize how the activation functions, bias and weights (linear transformations) serve to stretch the high dimensional space so that data go into extremes and become easy to put in a high dimension, low dimensional object (the manifold) where is trivial to classify or separate.

Gaining an intuition about this process will make some deep learning practices so much easy to understand.

behnamoh• 1 year ago

a few intuitions coming from theory (that was not topology).

I think these 'intuitions' are an after-the-fact thing, meaning AFTER deep learning comes up with a method, researchers in other fields of science notice the similarities between the deep learning approach and their (possibly decades old) methods. Here's an example where the author discovers that GPT is really the same computational problems he has solved in physics before:

https://ondrejcertik.com/blog/2023/03/fastgpt-faster-than-py...

ogogmad2• 1 year ago

I beg to differ. It's complete hyperbole to suggest that the article said "it's the same problem as something in physics", given this statement:

     It seems that the bottleneck algorithm in GPT-2 inference is matrix-matrix multiplication. For physicists like us, matrix-matrix multiplication is very familiar, *unlike other aspects of AI and ML* [emphasis mine]. Finding this familiar ground inspired us to approach GPT-2 like any other numerical computing problem.

Note: Matrix-matrix multiplication is basic mathematics, and not remotely interesting as physics. It's not physically interesting.

constantcrying2• 1 year ago

You are exactly right, after deep learning researchers had invented Adam for SGD, numerical analysts finally discovered Gradient descent. And after the first neural net was discovered, finally the matrix was invented in the novel field of linear algebra.

sota_pop2• 1 year ago

I disagree with this wholeheartedly. Sure, there is lots of trial and error, but it’s more an amalgamation of theory from many areas of mathematics including but not limited to: topology, geometry, game theory, calculus, and statistics. The very foundations (i.e. back-propagation) is just the chain rule applied to the weights. The difference is that deep learning has become such an accessible (sic profitable) field that many practitioners have the luxury of learning the subject without having to learn the origins of the formalisms. Ultimately allowing them to utilize or “reinvent” theories and techniques often without knowing they have been around in other fields for much longer.

kwertzzz• 1 year ago

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.

saberience• 1 year ago

None of the major aspects of deep learning came from manifolds though.

It is primarily linear algebra, calculus, probability theory and statistics, secondarily you could add something like information theory for ideas like entropy, loss functions etc.

But really, if "manifolds" had never been invented/conceptualized, we would still have deep learning now, it really made zero impact on the actual practical technology we are all using every day now.

niemandhier2• 1 year ago

It’s alchemy.

Deep learning in its current form relates to a hypothetical underlying theory as alchemy does to chemistry.

In a few hundred years the Inuktitut speaking high schoolers of the civilisation that comes after us will learn that this strange word “deep learning” is a left over from the lingua franca of yore.

adamnemecek2• 1 year ago

Not really, most of the current approaches are some approximations of the partition function.

Koshkin3• 1 year ago

Data doesn't actually live on a manifold.

Often, they do (and then they are called "sheaves").

wenc3• 1 year ago

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.

theahura16• 1 year ago

I say this as someone who has been in deep learning for over a decade now: this is pretty wrong, both on the merits (data obviously lives on a manifold) and on its applications to deep learning (cf chris olah's blog as an example from 2014, which is linked in my post -- https://colah.github.io/posts/2014-03-NN-Manifolds-Topology/). Embedding spaces are called 'spaces' for a reason. GANs, VAEs, contrastive losses -- all of these are about constructing vector manifolds that you can 'walk' to produce different kinds of data.

almostgotcaught• 1 year ago

You're citing a guy that never went to college (has no math or physics degree), has never published a paper, etc. I guess that actually tracks pretty well with how strong the whole "it's deep theory" claim is.

umutisik2• 1 year ago

If data did live on a manifold contained, e.g. images in R^{n^2}, then it wouldn't have thickness or branching, which it does. It's an imperfect approximation to help think about it. The use of mathematical language is not the same as an application of mathematics (and the use of the word 'space' there is not about topology).

esafak3• 1 year ago

It does if you relax your definition to accommodate approximation error, cf. e.g., Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning (https://aclanthology.org/2021.acl-long.568.pdf)

csimon80• 1 year ago

"All models are wrong, but some are useful" -George Box

woopwoop• 1 year ago

I don't agree with your first sentence, but I agree with the rest of this post.

Regic• 1 year ago

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...

nomel• 1 year ago

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.

HarHarVeryFunny2• 1 year ago

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.

baxtr• 1 year ago

Just a side comment to your observation: the principle is called reductionism and has been tried on many fields.

Physics is just applied mathematics

Chemistry is just applied physics

Biology is just applied chemistry

It doesn’t work very well.

yubblegum• 1 year ago

Near total majority, if not 100%, of the useful things done in deep learning have come from not thinking about topology in any way.

Of course. Now, to actually deeply understand what is happening with these constructs, we will use topology. Topoligical insights will without doubt then inform the next generations of this technology.

solomatov• 1 year ago

May I ask you to give examples of insights from topology which improved existing models, or at least improved our understanding of them? arxiv papers are preferred.

constantcrying2• 1 year ago

it's an empirical field advanced mostly by trial and error and, sure, a few intuitions coming from theory (that was not topology).

Neural Networks consist almost exclusively of two parts, numerical linear algebra and numerical optimization.

Even if you reject the abstract topological description. Numerical linear algebra and optimization couldn't be any more directly applicable.

sota_pop2• 1 year ago

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 polar |r,theta> basis). All things considered, nice article.

theahura16• 1 year ago

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)

benl• 1 year ago

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.

polotics• 1 year ago

Ok, how do transformers fit into this understanding of deep learning?

motoboi2• 1 year ago

Transformers don't feel differentiable (because of the attention mechanism), but they actually are (as being back-propagation based forces it to be).

The attention mechanism is not a stretching of the manifold, but is trained to be able to measure distances in the manifold surface, which is stretched and deformed (or transformed?) in the feed-forward layers.

theahura16• 1 year ago

Transformers learn embedding representations of tokens, which are easily mapped into a space. Similar tokens are mapped to similar places on the space. The fully connected layer at the end of each transformer block defines a transformation of a set of points in a space to another point in that space, not unlike the example of adding colors together to get a new color

jebarker2• 1 year ago

Transformers (with self-attention being the key operation) are kernel smoothers which fits easily into this view of the world. See here: http://bactra.org/notebooks/nn-attention-and-transformers.ht...

parpfish• 1 year ago

This is also how I've often thought about deep learning -- focusing on the geometry of the data at each layer rather than the weights and biases is far more revealing.

I've always been hopeful that some algebraic topology master would dig into this question and it'd provide some better design principles for neural nets. which activation functions? how much to fan in/out? how many layers?

profchemai2• 1 year ago

Once I read "This has been enough to get us to AGI.", credibility took a nose dive.

In general it's a nice idea, but the blogpost is very fluffy, especially once it connects it to reasoning, there is serious technical work in this area (i.g. https://arxiv.org/abs/1402.1869) that has expanded this idea and made it more concrete.

mirekrusin2• 1 year ago

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

ComplexSystems2• 1 year ago

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

mirekrusin2• 1 year ago

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

fedeb95• 1 year ago

Interesting read. This seems hard to prove:

"Everything lives on a manifold"

ComplexSystems2• 1 year ago

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 topology is not about: a notion of points being "close" or "far." If we have some topological space in which two points are "close," we can stretch the space so as to get the same topological space, but with the two points now "far". That's the whole point of the joke that the coffee cup and the donut are the same thing.

Instead, the entire thing seems to be a real-world application of something like algebraic geometry. We want to look for something like an algebraic variety the points are near. It's all about geometry and all about metrics between points. That's what it seems like to me, anyway.

srean12• 1 year ago

This is everything that topology is not about

100 percent true.

I can only hope that in an article that is about two things, i) topology and ii) deep learning, the evident confusions are contained within one of them -- topology, only.

theahura16• 1 year ago

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

srean12• 1 year ago

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.

steppi• 1 year ago

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

Koshkin3• 1 year ago

Speaking of: https://www2.math.upenn.edu/~ghrist/notes.html

soulofmischief2• 1 year ago

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 via learning a decontextualized pattern space on a given set of inputs. Given the inherent probabilistic nature of sampling, true reasoning is expressed in terms of probabilities, not axioms. It may be possible to discover axioms by locating fixed points or attractors on the manifold, but ultimately you're looking at a probabilistic manifold constructed from your input set.
But I don't think you can untie this "reasoning" from your input data. It's possible you will find "meta-reasoning", or similar structures found in any sufficiently advanced reasoning manifold, but these highly decontextualized structures might be entirely useless without proper recontextualization, necessitating that a reasoning manifold is trained on input whose patterns follow learnable underlying rules, if the manifold is to be useful for processing input of that kind.
Decontextualization is learning, decomposing aspects of an input into context-agnostic relationships. But recontextualization is the other half of that, knowing how to take highly abstract, sometimes inexpressible, context-agnostic relationships and transform them into useful analysis in novel domains

Full comment: https://news.ycombinator.com/item?id=42871894

mjburgess6• 1 year ago

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 propositions represented by them, since this is essential for correct reasoning -- and a relatively trivial capability to provide.

Eg., "if the spider is in boxA, then it is not everywhere else" and so on

jvanderbot3• 1 year ago

I suspect, as a layperson who watches people make decisions all the time, that somewhere in our mind is a "certainty checker".

We don't do logic itself, we just create logic from certainty as part of verbal reasoning. It's our messy internal inference of likelihoods that causes us to pause and think, or dash forward with confidence, and convincing others is the only place we need things like "theorems".

This is the only way I can square things like intuition, writing to formalize thoughts, verbal argument, etc, with the fact that people are just so mushy all the time.

soulofmischief2• 1 year ago

Propositions are just predictions, they all come with some level of uncertainty even if we ignore that uncertainty for practical purposes.

Any validation of a theory is inherently statistical, as you must sample your environment with some level of precision across spacetime, and that level of precision correlates to the known accuracy of hypotheses. In other words, we can create axiomatic systems of logic, but ultimately any attempt to compare them to reality involves empirical sampling.

Unlike classical physics, our current understanding of quantum physics essentially allows for anything to be "possible" at large enough spacetime scales, even if it is never actually "probable". For example, quantum tunneling, where a quantum system might suddenly overcome an energy barrier despite lacking the required energy.

Every day when I walk outside my door and step onto the ground, I am operating on a belief that gravity will work the same way every time, that I won't suddenly pass through the Earth's crust or float into the sky. We often take such things for granted, as axiomatic, but ultimately all of our reasoning is based on statistical correlations. There is the ever-minute possibility that gravity suddenly stops working as expected.

if the spider is in boxA, then it is not everywhere else

We can't even physically prove that. There's always some level of uncertainty which introduces probability into your reasoning. It's just convenient for us to say, "it's exceedingly unlikely in the entire age of the universe that a macroscopic spider will tunnel from Box A to Box B", and apply non-probabilistic heuristics.

It doesn't remove the probability, we just don't bother to consider it when making decisions because the energy required for accounting for such improbabilities outweighs the energy saved by not accounting for them.

As mentioned in my comment, there's also the possibility that universal axioms may be recoverable as fixed points in a reasoning manifold, or in some other transformation. If you view these probabilities as attractors on some surface, fixed points may represent "axioms" that are true or false under any contextual transformation.

naasking3• 1 year ago

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.

maxiepoo• 1 year ago

Isn't it more differential geometry?

srean12• 1 year ago

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 machine learning, -- scale, distance, angles etc., all usually provide lots of essential information about the data.

If you want to distinguish between a tabby cat and a tiger it would be an act of stupidity to ignore scale.

Topology is useful especially when you cannot trust lengths, distances angles and arbitrary deformations. That happens, but to claim deep learning is applied topology is absurd, almost stupid.

theahura16• 1 year ago

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

srean12• 1 year ago

But...you can't.

Only in a desperate sales pitch or a desparate research grants. There are of course some situations were certain measurements are untrustworthy, but to claim that is the common case is very snake oily.

When certain measurements become untrustworthy, that it does so only because of some unknown smooth transformation, is not very frequent (this is what purely topological methods will deal with). Random noise will also do that for you.

Not disputing the fact that sometimes metrics cannot be trusted entirely, but to go to a topological approach seems extreme. One should use as much of the relevant non-topological information as possible.

As the hackneyed example goes a topological methods would not be able to distinguish between a cup and a donut. For that you would need to trust non-topological features such as distances and angles. Deep learning methods can indeed differentiate between cop-nip and coffee mugs.

BTW I am completely on-board with the idea that data often looks as if it has been sampled from an unknown, potentially smooth, possibly non-Euclidean manifold and then corrupted by noise. In such cases recovering that manifold from noisy data is a very worthy cause.

In fact that is what most of your blogpost is about. But that's differential geometry and manifolds, they have structure far richer than a topology. For example they may have tangent planes, a Reimann metric or a symplectic form etc. A topological method would throw all of that away and focus on topology.

kentuckyrobby• 1 year ago

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.

throwawaymaths• 1 year ago

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

quantadev5• 1 year ago

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.

srean12• 1 year ago

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.

cvoss• 1 year ago

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.

terabytest• 1 year ago

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 intelligence (the closest thing we know to AGI) requires extremely complex sensory and internal feedback loops and continuous processing unlike autoregressive models' discrete processing.

As a layman, this matches my intuition that LLMs are not at all in the same family of systems as the ones capable of generating intelligence or consciousness.

theahura16• 1 year ago

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 networks and biological networks are actually quite similar. For example, in mice olfactory systems there is something akin to a 'feature vector' that appears based on which neurons light up. Specific sets of neurons lighting up means 'chocolate' or 'lemon' or whatever. More generally, it seems like neuronal representations are somewhat similar to embedding representations, and you could imagine constructing an embedding space based on what neurons light up where. Everything on top of the embeddings is 'just' processing.

fusionadvocate• 1 year ago

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.

_alternator_• 1 year ago

The question is not so much whether this is true—we can certainly represent any data as points on a manifold. Rather, it’s the extent to which this point of view is useful. In my experience, it’s not the most powerful perspective.

In short, direct manifold learning is not really tractable as an algorithmic approach. The most powerful set of tools and theoretical basis for AI has sprung from statistical optimization theory (SGD, information-theoretical loss minimization, etc.). The fact that data is on a manifold is a tautological footnote to this approach.

crgi• 1 year ago

Interesting read. Reminded me of the Trinity 3D manifold visualization tool which (among other things) let's you explore the hyperspace of neural networks: https://github.com/trinity-xai/Trinity

vayllon• 1 year ago

Another type of topology you’ll encounter in deep neural networks (DNNs) is network topology. This refers to the structure of the network — how the nodes are connected and how data flows between them. We already have several well-known examples, such as auto-encoders, convolutional neural networks (CNNs), and generative adversarial networks (GANs), all of which are bio-inspired.

However, we still have much to learn about the topology of the brain and its functional connectivity. In the coming years, we are likely to discover new architectures — both internal within individual layers/nodes and in the ways specialized networks connect and interact with each other.

Additionally, the brain doesn’t rely on a single network, but rather on several ones — often referred to as the "Big 7" — that operate in parallel and are deeply interconnected. Some of these include the Default Mode Network (DMN), the Central Executive Network (CEN) or the Limbic Network, among others. In fact, a single neuron can be part of multiple networks, each serving different functions.

We have not yet been able to fully replicate this complexity in artificial systems, and there is still much to be learned and inspired by from this "network topologies".

So, "Topology is all you need" :-)

mbowcut2• 1 year ago

To a topologist, everything is topology.

kookamamie2• 1 year ago

To a man with only a hammer, everything looks like a nail.

Graviscalar3• 1 year ago

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.

rubitxxx8• 1 year ago

What would you have expected to happen?

Advances and insights sometimes lie dormant for decades or more before someone else picks them up and does something new.

Graviscalar3• 1 year ago

I would expect model/algorithm improvements from using topological concepts to analyze the manifolds in question or concrete results in model interpretability. Gunnar has studied some toy examples, but they were barely a step up from the ones Olah constructed for the sake of explanation and they haven't borne any further fruit.

You can say any advance or insight is just lying dormant, it doesn't mean anything unless you can specifically articulate why it still has potential. I haven't made any claims on the future of the intersection of deep learning and topology, I was pointing out that it's been anything but dormant given the interest in it but it hasn't lead anywhere.

theahura16• 1 year ago

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.

Graviscalar3• 1 year ago

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.

jxramos• 1 year ago

Ayasdi immediately came to mind too seeing this post. I haven't thought of them in a long time, looks like they got bought out in 2019, prepandemic too which was probably best since mid pandemic had a lot of poor valuations

https://www.symphonyai.com/news/financial-services/ayasdi-jo...

adsharma• 1 year ago

Isn't Deep Learning more like Graph Theory? I shared yesterday that Google published a paper called CRISP (https://arxiv.org/pdf/2505.11471) that carefully avoids any reference to the word "Graph".

So then the question becomes what's the difference between Graph Theory and Applied Topology? Graphs operate on discrete structures and topology is about a continuous space. Otherwise they're very closely related.

But the higher order bit is that AI/ML and Deep Learning in particular could do a better job of learning from and acknowledging prior art from related fields. Reusing older terminology instead of inventing new.

kookamamie2• 1 year ago

This has been enough to get us to AGI.1

Hard disagree.

colah37• 1 year ago

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 perspective useful, but after a decade of grappling with trying to understand what goes on inside neural networks, I just haven't gotten that much traction out of it.

I've had a lot more success with:

* The linear representation hypothesis - The idea that "concepts" (features) correspond to directions in neural networks.

* The idea of circuits - networks of such connected concepts.

Some selected related writing:

- https://distill.pub/2020/circuits/zoom-in/

- https://transformer-circuits.pub/2022/mech-interp-essay/inde...

- https://transformer-circuits.pub/2025/attribution-graphs/bio...

montebicyclelo• 1 year ago

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

colah37• 1 year ago

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

agentcoops• 1 year ago

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.

winwang• 1 year ago

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

riemannzeta2• 1 year ago

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.

colah37• 1 year ago

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

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

theahura16• 1 year ago

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.

colah37• 1 year ago

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.

rajnathani• 1 year ago

I was going to comment the same about the Superposition hypothesis [0], when the OP comment (edit: Update: The OP commenter is (as pointed by other HN comments, the cofounder of Anthropic) behind the Superposition research) mentioned about "I've had a lot more success with: * The linear representation hypothesis - The idea that "concepts" (features) correspond to directions in neural networks", as this concept-per-NN-feature idea seems too "basic" to explain some of the learning which NNs can do on datasets. On one of our custom trained neural network models (not LLM, but audio-based and currently proprietary) we noticed the same of the ML model being able to "overfit" on a large amount of data despite not many few parameters relative to the size of the dataset (and that too with dropout in early layers).

[0] https://www.anthropic.com/research/superposition-memorizatio...

iNic• 1 year ago

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

godelski• 1 year ago

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

dang• 1 year ago

That earlier post had a few small HN discussions (for those interested):

Neural Networks, Manifolds, and Topology (2014) - https://news.ycombinator.com/item?id=19132702 - Feb 2019 (25 comments)

Neural Networks, Manifolds, and Topology (2014) - https://news.ycombinator.com/item?id=9814114 - July 2015 (7 comments)

Neural Networks, Manifolds, and Topology - https://news.ycombinator.com/item?id=7557964 - April 2014 (29 comments)

j2kun• 1 year ago

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?

3abiton• 1 year ago

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

colah37• 1 year ago

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.

adamnemecek2• 1 year ago

Consider looking into fields related to machine learning to see how topology is used there. The main problem is that some of the cool math did not survive the transition to CS, e.g. the math for control theory is not quite present in RL.

In terms of topology, control theory has some very cool topological interpretations, e.g. toruses appear quite a bit in control theory.

nis0s• 1 year ago

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.

khoangothe• 1 year ago

Cool post! Thanks

deepburner• 1 year ago

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.

ada1981• 1 year ago

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.

revskill• 1 year ago

So ml is about drawing the line.

Koshkin3• 1 year ago

On the TDA in general, I found this series of articles by Brandon Brown helpful:

https://github.com/outlace/outlace.github.io

CliffStoll• 1 year ago

Applied topology. Might a Klein bottle actually be a useful?

colah37• 1 year ago

It's a bit different than what's discussed here, but color-contrast detectors in neural networks can be thought of as forming a Klein bottle: https://distill.pub/2020/circuits/equivariance/#hue-rotation...

(This is, in some sense, for similar reason to Gunnar Carlson et al finding a Klein bottle when looking at high-contrast image patches, except one level more abstract, since it's about features rather than data points.)

garganzol• 1 year ago

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.

strimp099• 1 year ago

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.

paraschopra• 1 year ago

Aren't manifolds generally task-dependent?

I've been debating whether the data lies on a manifold, or whether the data attributes that are task-relevant (and of our interest) lie on a manifold?

I suspect it is the latter, but I've seen Platonic Representation Hypothesis that seems to hint it is the former.

xchip• 1 year ago

The deep learning you are talking about is better understood as a regression.

wg0• 1 year ago

This is an insightful read but at a place where a prompt churns out a whole model - that's yet far far away and extremely compute intensive.

Philosophical point being - if human brain is also all planes all the way down with trillion dimensional planes separating concepts.

I don't think so that is the case but yes - we might mimic something like that and such a thing will always malfunction more than their human counterparts in a given subject area and would always hit a wall where a human won't.

iFire• 1 year ago

Huge fan of the idea in fiction of Applied Topology is magic https://en.wikipedia.org/wiki/The_Laundry_Files

Howard is recruited to work for the Q-Division of SOE, otherwise known as "the Laundry", the British government agency which deals with occult threats. "Magic" is described as being a branch of applied computation (mathematics), therefore computers and equations are just as useful, and perhaps more potent, than classic spellbooks, pentagrams, and sigils for the purpose of influencing ancient powers and opening gates to other dimensions.

lesostep2• 1 year ago

as long as we can separate good from bad we can train a neural network to sort out the topology for us.

10-ish years ago, I saw a project training networks to guess biological sex from face photos. They carefully removed makeup, moustache, hair, etc, so the model would be unbiased, yet they only reached 70 to 80% correct guesses. Yet it seemed like a great result, and they were trying to reach 99%.

First thing I did after reading their paper, was seek a paper where people would try and guess the biological sex from similar photos. And people weren't that much better at it. The difference between people and machine guessing was 1 or 2 percent.

I asked the guys that run the project, how they proved that such a division, based only on a photo, was even possible. They didn't understand the question, they just assumed that you can do it.

They couldn't improve their results in the end. Maybe they sucked at teaching neural networks, or maybe a lot of faces just are unisex if you remove gender markers.

I bring this anecdote because this guys, in my eyes, made a reasonable assumption. An assumption that since they in most situations can guess what's in someone pants by seeing someones face, the face has this information.

The assumption that we could somehow separate good from bad, when we rewrite school books every year, when we try to calculate "half-life of knowledge", when philosophy as a science isn't over, and every day there are political and ideological debates about what's best, is a very-very unreasonable assumption.

lesostep2• 1 year ago

I forgot to conclude:

In the end, it's not even reasonable to assume that such a divide between "good" and "bad" exists at all.

Deep Learning Is Applied Topology

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