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:
> 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.
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).
Many mathematicians are (rightly, IMO) allergic to assertions that certain branches are not useful (explicit in OP) and especially so if they are dismissive of attempts to understand complicated real world phenomema (implicit in OP, if you ask me).
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.
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.
This is way too rigorous. You can absolutely have theories that lead to useful results even if they aren't as predictive as you describe. Theory of evolution for an obvious counterpoint.
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.
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.
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.
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.
> 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:
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.
Although, to try to see it from the author’s perspective, it is pulling tools out of the same (extremely well developed and studied in it’s own right) toolbox as computational physics does. It is a little funny although not too surprising that a computational physics guy would look at some linear algebra code and immediately see the similarity.
Edit: actually, thinking a little more, it is basically absurd to believe that somebody has had a career in computational physics without knowing they are relying heavily on the HPC/scientific computing/numerical linear algebra toolbox. So, I think they are just using that to help with the narrative for the blog post.
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.
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.
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.
Bethe ansatz is one. It took a toure de force by Yedidia to recognize that loopy belief propagation is computing the stationary point of Bethe's approximation to Free Energy.
Many statistical thermodynamics ideas were reinvented in ML.
Same is true for mirror descent. It was independently discovered by Warmuth and his students as Bregman divergence proximal minimization, or as a special case would have it, exponential gradient algorithms.
The connections of deep learning to stat-mech and thermodynamics are really cool.
It's led me to wonder about the origin of the probability distributions in stat-mech. Physical randomness is mostly a fiction (outside maybe quantum mechanics) so probability theory must be a convenient fiction. But objectively speaking, where then do the probabilities in stat-mech come from? So far, I've noticed that the (generalised) Boltzmann distribution serves as the bridge between probability theory and thermodynamics: It lets us take non-probabilistic physics and invent probabilities in a useful way.
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 temporal terms - when the macro-states are not changing with time.
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.
One might add 8-16-bit training and quantization. Also, computing semi-unreliable values with error correction. Such tricks have been used in embedded, software development on MCU's for some time.
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.
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.
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.
The reason deep learning is alchemy is that none of these deep theories have predictive ability.
Essentially all practical models are discovered by trial and error and then "explained" after the fact. In many papers you read a few paragraphs of derivation followed by a simpler formulation that "works better in practice". E.g., diffusion models: here's how to invert the forward diffusion process, but actually we don't use this, because gradient descent on the inverse log likelihood works better. For bonus points the paper might come up with an impressive name for the simple thing.
In most other fields you would not get away with this. Your reviewers would point this out and you'd have to reformulate the paper as an experience report, perhaps with a section about "preliminary progress towards theoretical understanding". If your theory doesn't match what you do in practice - and indeed many random approaches will kind of work (!) - then it's not a good theory.
It's true that there is no directly predictive model of deep learning, and it's also true that there is some trial and error, but it is wrong to say that therefore there is no operating theory at all. I recommend reading Ilyas 30 papers (here's my review of that set:
https://open.substack.com/pub/theahura/p/ilyas-30-papers-to-...) to see how shared intuitions and common threads are clearly developed over the last decade+
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.
Any reasonable statistical explanation of deep learning requires there to be some sort of low dimensional latent structure in the data. Otherwise, we would not have enough training data to learn good models, given how high the ambient dimensions are for most problems.
It turns out a lot of disconnected spaces can be approximated by smooth ones that have really sharp boundaries, which more or less seems to be how neural networks will approximate something like discrete tokens
Can be approximated yes. Approximated well? No, but you can get away with it sometimes with saturation functions like softmax. But badly. It’s like trying to solve an integer program as a linear program. You end up with a relaxation that is not integral and not the real answer.
An integer lattice can only be a manifold in a trivial sense (dimension 0). But not for any positive dimensions.
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.
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.
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).
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)
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...
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.
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.
> 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.
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.
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.
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.
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.
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
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?
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.
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.
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.
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.
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.
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
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
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.
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, it disappears. It's exactly and only a project of the child playing with his lego, certain that great engineering projects have little use for any more than stacking bricks.
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 from a logical deduction.
Something more like probability factor graph can do it, without the pernicious use of "concepts" or too much need for implication, which is sticky and overly rigorous.
That's all I have to say, and I doubt we'll find middle ground.
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.
Hey it's possible you're absolutely right, but our arguments have about equal value (none) because they are both presented as personal opinions with zero external support. I just had the decency to admit that up front.
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.
This response doesn't fill me with confidence. You aren't really engaging with any of the actual issues your position entails.
A proposition is not a prediction. A prediction is either an estimate of the value of some quantity ("the dumb ML meaning of prediction") or a proposition which describes a future scenario. We can trivially enumerate propositions that do not describe future scenarios, eg., 2 + 2 = 4.
Uncertainty is a property of belief attitudes towards propositions, it isn't a feature of their semantic content. A person doesnt mean anything different by "2 + 2 = 4" if they are 80 or 90% sure of it.
> We can't even physically prove that.
Irrelevant. Our minds are not constrained by physical possibility, necessarily so, as we know very little about what is physically possible. I can imagine abitary number of cases, arising out of logical manipulation of propositons, that are not physically possible. (Eg., "Superman can lift any building. The empire state building is so-and-so a kind of building. Imagine(Superman lifting the empire state building)").
The infinite variety of our imagination is a trivial consequence of non-probabilistic operations on propositions, it's incomprehensibly implausible as a consequence of merely probabilistic ones.
That nature seems to have endowed minds with discrete operations, that these are empirical in operation across very wide classes of reasoning, including imagination, that these seem trivial for nature to provide (etc.) render the notion that they don't exist highly highly implausible.
There is nothing lacking explanation here. The relevant mental processes we have to hand are fairly obvious and fairly easy to explain.
Its an obvious act of credulity to try and find some way to make the latest trinkets of the recent rich some sort of miracle. All of these
projects of "incredible abstraction" follow around these hype cycles, turning lead into gold: if x "is really" y, and y "is really" z, and ..., then x is amazin! This piles towers of every more general hollowed-out words on top of each other until the most trivial thing sounds like a wonder.
> It's highly implausible animals would have been endowed with no ability to operate non-probabilistically on propositions represented by them, since this is essential for correct reasoning
Why would animals need to evolve 100% correct reasoning if probabilistically correct reasoning suffices? If probabilistic reasoning is cheaper in terms of energy then correct reasoning is a disadvantage.
It doesnt suffice. It's also vastly energetically cheaper just to have (algorithmic) negation. Compressing (A, not A) into a probability function is extremely incomprehensibly expensive.
> 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.
> It is a matter of empirical fact that we can reason with logical relationships
It is a matter of empirical fact that our ability to correctly reason with logical relationships only has high statistical likelihood, not certainty. This looks less like actual logic and more like a probabilistic model of logic.
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.
> 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
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.
I don't think that was their point, I think their point was that neural networks 'create' their optimization space by using lengths, distances, and angles. You can't reframe it from a topological standpoint, otherwise optimization spaces of some similar neural networks on similar problems would topologically comparable, which is not true.
Well, sorta. There is some evidence to suggest that neural networks learn 'universal' features (cf Anthropic's circuits thread). But I'll openly admit to being out of my depth here, and maybe I just don't understand OPs point
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
The word "topology" has a legitimate dictionary definition, that has none of the requirements that you're asserting. I think what you're missing is that it has two definitions.
In blog posts about specialised and technical topics it is expected that in-domain technical keywords that have long established definitions and meanings be used in the same technical sense. Otherwise it can become quite confusing. Gravity means gravity when we are talking Newtonian mechanics. Similarly, in math and ML 'topology' has a specific meaning.
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 appear in the same text often.
Topology is a traditional as well as an active branch of applied and pure mathematics, well, Physics too.
Math magazines for high schoolers have articles on it. Colleges offer multiple courses on it. Some of those courses would be mandatory for a degree in even undergrad mathematics.
If one wants to do graduate studies then one can do a Masters or a PhD in Topology, well in one of it's many branches.
It's also not a new kid on the block. It goes back to ... analysis situs ... further back to Leibniz, although it began to crystalize formally after Poincare.
If someone wants to use the phrase 'differential calculus' to mean something else in their love letters and sweet nothings, that's absolutely fine :) but in Maths (and Machine Learning, well, with quality of peer reviewing this might soon be iffy) it has a well established and unambiguous meaning.
Note because of its shared beginning at the feet of Leibniz, comparing it with calculus is not an unfair comparison.
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.
"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.
The question was never "Is topology a field of mathematics". The question was, is that term most often used to refer to surfaces in general, and the answer to that is still 'yes'.
Could you drop Amazon a message. They should really change the search results if that's what topology means in general. I am sure they would be delighted to receive the bug report.
For laughs, I asked ChatGPT to use 'topology' in place of 'surface'. Here's what it wrote:
The topology of the lake was so calm it reflected the mountains perfectly.
She wiped the kitchen topology clean after cooking dinner.
After years of silence, the truth began to topology.
The spacecraft landed safely on the topology of Mars.
A thin layer of dust had settled on the topology of the old table.
He barely scratched the topology of the topic in his presentation.
As the submarine ascended, it broke through the topology of the ocean.
The topology of the road was slick with ice.
Despite her calm topology, she was extremely nervous inside.
The paint bubbled and peeled off the topology due to the heat.
Our disagreement aside, I think these are hilarious. We should agree on that.
The best was
The surface of a doughnut resembles that of a coffee mug due to their similar structure.
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.
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.
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.
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.
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.
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
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".
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.
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.
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.
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.
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
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.
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[ 3.2 ms ] story [ 221 ms ] threadTopological 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)
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.
Even if existing theory is inadequate, would an operating theory not be beneficial?
Or is the mystique combined with guess&check drudgery job security?
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.
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.
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.
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...
Although, to try to see it from the author’s perspective, it is pulling tools out of the same (extremely well developed and studied in it’s own right) toolbox as computational physics does. It is a little funny although not too surprising that a computational physics guy would look at some linear algebra code and immediately see the similarity.
Edit: actually, thinking a little more, it is basically absurd to believe that somebody has had a career in computational physics without knowing they are relying heavily on the HPC/scientific computing/numerical linear algebra toolbox. So, I think they are just using that to help with the narrative for the blog post.
Many statistical thermodynamics ideas were reinvented in ML.
Same is true for mirror descent. It was independently discovered by Warmuth and his students as Bregman divergence proximal minimization, or as a special case would have it, exponential gradient algorithms.
One can keep going.
It's led me to wonder about the origin of the probability distributions in stat-mech. Physical randomness is mostly a fiction (outside maybe quantum mechanics) so probability theory must be a convenient fiction. But objectively speaking, where then do the probabilities in stat-mech come from? So far, I've noticed that the (generalised) Boltzmann distribution serves as the bridge between probability theory and thermodynamics: It lets us take non-probabilistic physics and invent probabilities in a useful way.
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 temporal terms - when the macro-states are not changing with time.
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.
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.
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.
Essentially all practical models are discovered by trial and error and then "explained" after the fact. In many papers you read a few paragraphs of derivation followed by a simpler formulation that "works better in practice". E.g., diffusion models: here's how to invert the forward diffusion process, but actually we don't use this, because gradient descent on the inverse log likelihood works better. For bonus points the paper might come up with an impressive name for the simple thing.
In most other fields you would not get away with this. Your reviewers would point this out and you'd have to reformulate the paper as an experience report, perhaps with a section about "preliminary progress towards theoretical understanding". If your theory doesn't match what you do in practice - and indeed many random approaches will kind of work (!) - then it's not a good theory.
Often, they do (and then they are called "sheaves").
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.
An integer lattice can only be a manifold in a trivial sense (dimension 0). But not for any positive dimensions.
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.
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.
Physics is just applied mathematics
Chemistry is just applied physics
Biology is just applied chemistry
It doesn’t work very well.
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.
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.
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.
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?
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.
"Everything lives on a manifold"
"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.
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.
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.
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
> 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
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
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.
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, it disappears. It's exactly and only a project of the child playing with his lego, certain that great engineering projects have little use for any more than stacking bricks.
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 from a logical deduction.
Something more like probability factor graph can do it, without the pernicious use of "concepts" or too much need for implication, which is sticky and overly rigorous.
That's all I have to say, and I doubt we'll find middle ground.
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.
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.
A proposition is not a prediction. A prediction is either an estimate of the value of some quantity ("the dumb ML meaning of prediction") or a proposition which describes a future scenario. We can trivially enumerate propositions that do not describe future scenarios, eg., 2 + 2 = 4.
Uncertainty is a property of belief attitudes towards propositions, it isn't a feature of their semantic content. A person doesnt mean anything different by "2 + 2 = 4" if they are 80 or 90% sure of it.
> We can't even physically prove that.
Irrelevant. Our minds are not constrained by physical possibility, necessarily so, as we know very little about what is physically possible. I can imagine abitary number of cases, arising out of logical manipulation of propositons, that are not physically possible. (Eg., "Superman can lift any building. The empire state building is so-and-so a kind of building. Imagine(Superman lifting the empire state building)").
The infinite variety of our imagination is a trivial consequence of non-probabilistic operations on propositions, it's incomprehensibly implausible as a consequence of merely probabilistic ones.
That nature seems to have endowed minds with discrete operations, that these are empirical in operation across very wide classes of reasoning, including imagination, that these seem trivial for nature to provide (etc.) render the notion that they don't exist highly highly implausible.
There is nothing lacking explanation here. The relevant mental processes we have to hand are fairly obvious and fairly easy to explain.
Its an obvious act of credulity to try and find some way to make the latest trinkets of the recent rich some sort of miracle. All of these projects of "incredible abstraction" follow around these hype cycles, turning lead into gold: if x "is really" y, and y "is really" z, and ..., then x is amazin! This piles towers of every more general hollowed-out words on top of each other until the most trivial thing sounds like a wonder.
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.
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 should be obviously doubly so, since as a model -- as you point out -- it makes trivial aspects of our cognition impossibly expensive to acqurie.
It is a matter of empirical fact that our ability to correctly reason with logical relationships only has high statistical likelihood, not certainty. This looks less like actual logic and more like a probabilistic model of logic.
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.
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
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.
Dogs have fur. Dogs are an example of a furry animal. But dogs and furs are not the same thing although they may appear in the same text often.
Topology is a traditional as well as an active branch of applied and pure mathematics, well, Physics too.
It has tons of text books printed on it, has several active journals and conferences dealing with it. https://www.amazon.com/s?k=Topology&sprefix=topology+%2Caps%...
Surprise, surprise ...not ...has an extensive Wikipedia page.
https://en.m.wikipedia.org/wiki/Topology
Math magazines for high schoolers have articles on it. Colleges offer multiple courses on it. Some of those courses would be mandatory for a degree in even undergrad mathematics.
If one wants to do graduate studies then one can do a Masters or a PhD in Topology, well in one of it's many branches.
It's also not a new kid on the block. It goes back to ... analysis situs ... further back to Leibniz, although it began to crystalize formally after Poincare.
If someone wants to use the phrase 'differential calculus' to mean something else in their love letters and sweet nothings, that's absolutely fine :) but in Maths (and Machine Learning, well, with quality of peer reviewing this might soon be iffy) it has a well established and unambiguous meaning.
Note because of its shared beginning at the feet of Leibniz, comparing it with calculus is not an unfair comparison.
(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.
Could you drop Amazon a message. They should really change the search results if that's what topology means in general. I am sure they would be delighted to receive the bug report.
For laughs, I asked ChatGPT to use 'topology' in place of 'surface'. Here's what it wrote:
Our disagreement aside, I think these are hilarious. We should agree on that.The best was
This applies both ways. Maybe you could relax with the facetiousness? It doesn’t help your argument and makes you look like an ass.
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.
> 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.
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.
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" :-)
Advances and insights sometimes lie dormant for decades or more before someone else picks them up and does something new.
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.
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.
https://www.symphonyai.com/news/financial-services/ayasdi-jo...
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.