The big questions I’m taking away are:

(1) they are claiming to produce apparently bijective closed-form symbolic representations/approximations of, among other things, LLMs. Is evaluating these closed-form representations more computationally efficient? The implications of that are potentially huge. It would be essentially analytic distillation. Fable on a chip and not a data center would be important — and disruptive - in many ways.

(2) Unsupervised, and even supervised, symbolic approaches to problem solving break down due to combinatorial explosion, among other things. This could potentially allow us to treat LLM training and inference as a search algorithm for novel symbolic approaches to solving new classes of complex problems hitherto unreachable through other approaches. If that works, I suspect it’s a feedback loop, too - the learnings from one representation push advances in the other. This would also increase the economic value of large training runs, since the model itself is now valuable, not just its inference.

(3) Per the above, can this push LLM design to greater capabilities?

The relationship between this and Anthropic’s J-space observation is also interesting. This is much, much deeper and more directly actionable, though.

EDIT: I ran my questions through Sonnet — yes, I appreciate the irony — and it was none too sanguine about questions (1) and (2), but thought (3) was reasonable. In any case, this is quite the paper. On reflection, I do think that the apparent reliance on very simple symbolic representations and tasks is underwhelming. But the approach is impressive. And obviously this is still early days, and the value of building a bridge between the very fuzzy LLM models and the rigorous, mechanically provable models would be enormous.

The math and core experimentation here is beyond my abilities, but what I think I understand is that there are possible deeper patterns of representation that exist in LLMs that are distillations of core conceptual relations in grammar that we can get our heads around in a mathematical sense rather than apparent layer-smeared noise that somehow, un-interpretably (in a meaningful sense), resolve to correct grammar/inferences.

That's pretty cool. I hope I've got that kinda-right.

Sounds reasonable... That the model is sometimes learning a lossy vector representation of something symbolic in nature... Sure, a NN can approximate a function?

They say this holds in... Some examples they found?

I don't enough about this area

The human mind cannot comprehend the capacity of massively multidimensional space.

Just going from 2D to 3D creates massive new positional potential (e.g. surface of the earth, vs. the atmosphere above earth...).

Now imagine 1,000 dimensions.

this is an obvious result. for example, this guy has been writing on substack about this for at least a year or two (with code snippets) explaining the phenomenon of grokking and the ghostbasin.com concept - https://richardaragon.substack.com/

their algorithm is even named "DISCOVER" so they set out to discover the connective tissue of why the universe has invariants like math, and lo it was discovered.

i guess good job for having credentials & publishing the math so people 2years behind the curve can learn from your tenure?

yes. large matrices can gradient descend to understand arbitrary symbolic logic.

ENGLISH IS INSUFFICIENT but it is at least a few decades of math proofs & progress :) welcome to the future Slackernews

It's like Neo says "You get used to it, though. Your brain does the translating. I don't even see the code." He was referring to something like a K, Q, V vector at the time I believe.

"Vectors seem inadequate for capturing the structure of language, logic, and other cognitive domains, yet neural networks achieve impressive performance in these areas". Missing the forest for the trees? Aren't neural networks modeled after biological systems? Our brains are obviously able to contain symbolic structure despite not having a "symbol processing unit".