Deep Learning Is Applied Topology

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.

O neat, I haven't read that far back. Will add it to the reading list.

To flesh this out a bit, part of why I find circuits less compelling is because it seems intuitive to me that neural networks more or less smoothly blend 'process' and 'state'. As an intuition pump, a vector x matrix matmul in an MLP can be viewed as changing the basis of an input vector (ie the weights act as a process) or as a way to select specific pieces of information from a set of embedding rows (ie the weights act as state).

There are architectures that try to separate these out with varying degrees of success -- LSTMs and ResNets seem to have a more clear throughline of 'state' with various 'operations' that are applied to that state in sequence. But that seems really architecture-dependent.

I will openly admit though that I am very willing to be convinced by the circuits paradigm. I have a background in molecular bio and there's something very 'protein pathways' about it.

> Linear representations aren't necessarily in tension with the manifold hypothesis.

True! I suppose I was thinking about a 'strong' form of linear representations, which is something like: features are represented by linear combinations of neurons that display the same repulsion-geometries as observed in Toy Models, but that's not what you're saying / that's me jumping a step too far.

> GANs/VAEs/etc modeling things as a latent gaussian space is actually way more natural if you allow superposition

Superposition is one of those things that has always been so intuitive to me that I can't imagine it not being a part of neural network learning.

But I want to make sure I'm getting my terminology right -- why does superposition necessarily require the linear representation hypothesis? Or, to be more specific, does [individual neurons being used in combination with other neurons to represent more features than neurons] necessarily require [features are linear compositions of neurons]?

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.

Note this happens in "uniform superposition". In reality, we're almost certainly in very non-uniform superposition.

One key term to look for is "feature manifolds" or "multi-diemsnional features". Some discussion here: https://transformer-circuits.pub/2024/july-update/index.html...

(Note that the term "strong linear representation" is becoming a term of art in the literature referring to the idea that all features are linear, rather than just most or some.)

> I want to make sure I'm getting my terminology right -- why does superposition necessarily require the linear representation hypothesis? Or, to be more specific, does [individual neurons being used in combination with other neurons to represent more features than neurons] necessarily require [features are linear compositions of neurons]?

When you say "individual neurons being used in combination with other neurons to represent more features than neurons", that's a way one might _informally_ talk about superposition, but doesn't quite capture the technical nuance. So it's hard to know the full scope of what you intend. All kinds of crazy things are possible if you allow non-linear features, and it's not necessarily clear what a feature would mean.

Superposition, in the narrow technical sense of exploiting compressed sensing / high-dimensional spaces, requires linear representations and sparsity.