Deep Learning Is Applied Topology

> 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.

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

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

And crucially, the initialization algorithm.

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

> 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.

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.

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.

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.

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.

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.

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.

> 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.

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.

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

I 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.

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.

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.

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

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.

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.

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.

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.