Language Models Pack Billions of Concepts into 12,000 Dimensions

I think the author is too focused on the case where all vectors are orthogonal and as a consequence overestimates the amount of error that would be acceptable in practice. The challenge isn't keeping orthogonal vectors almost orthogonal, but keeping the distance ordering between vectors that are far from orthogonal. Even much smaller values of epsilon can give you trouble there.

So the claim that "This research suggests that current embedding dimensions (1,000-20,000) provide more than adequate capacity for representing human knowledge and reasoning." is way too optimistic in my opinion.

Tangential, but the ChatGPT vibe of most of the article is very distracting and annoying. And I say this as someone who consistently uses AI to refine my English. However, I try to avoid letting it reformulate too dramatically, asking it specifically to only fix grammar and non-idiomatic parts while keeping the tone and formulation as much as possible.

Beyond that, this mathematical observation is genuinely fascinating. It points to a crucial insight into how large language models and other AI systems function. By delving into the way high-dimensional data can be projected into lower-dimensional spaces while preserving its structure, we see a crucial mechanism that allows these models to operate efficiently and scale effectively.

These set of intuitions and the Johnson-Lindenstrauss lemma in particular are what power a lot of the research effort behind SAEs (Sparse Autoencoders) in the field of mechanistic interpretability in AI safety.

A lot of the ideas are explored in more detail in Anthropic's 2022 paper that's one of the foundational papers in SAE research: https://transformer-circuits.pub/2022/toy_model/index.html

My intuition of this problem is much simpler — assuming there’s some rough hierarchy of concepts, you can guesstimate how many concepts can exist in a 12,000-d space by taking the combinatorial of the number of dimensions. In that world, each concept is mutually orthgonal with every other concept in at least some dimension. While that doesn’t mean their cosine distance is large, it does mean you’re guaranteed a function that can linearly separate the two concepts.

It means you get 12,000! (Factorial) concepts in the limit case, more than enough room to fit a taxonomy

What's the point of the relu in the loss function? Its inputs are nonnegative anyway.

Wow, I think I might just have grasped one of the sources of the problems we keep seeing with LLMs.

Johnson-Lichtenstrauss guarantees a distance preserving embedding for a finite set of points into a space with a dimension based on the number of points.

It does not say anything about preserving the underlying topology of the contious high dimensional manifold, that would be Takens/Whitney-style embedding results (and Sauer–Yorke for attractors).

The embedding dimensions needed to fulfil Takens are related to the original manifolds dimension and not the number of points.

It’s quite probable that we observe violations of topological features of the original manifold, when using our to low dimensional embedded version to interpolate.

I used AI to sort the hodge pudge of math in my head into something another human could understand, edited result is below:

=== AI in use === If you want to resolve an attractor down to a spatial scale rho, you need about n ≈ C * rho^(-d_B) sample points (here d_B is the box-counting/fractal dimension).

The Johnson–Lindenstrauss (JL) lemma says that to preserve all pairwise distances among n points within a factor 1±ε, you need a target dimension

k ≳ (d_B / ε^2) * log(C / rho).

So as you ask for finer resolution (rho → 0), the required k must grow. If you keep k fixed (i.e., you embed into a dimension that’s too low), there is a smallest resolvable scale

rho* (roughly rho* ≳ C * exp(-(ε^2/d_B) * k), up to constants),

below which you can’t keep all distances separated: points that are far on the true attractor will show up close after projection. That’s called “folding” and might be the source of some of the problems we observe .

=== AI end ===

Bottom line: JL protects distance geometry for a finite sample at a chosen resolution; if you push the resolution finer without increasing k, collisions are inevitable. This is perfectly consistent with the embedding theorems for dynamical systems, which require higher dimensions to get a globally one-to-one (no-folds) representation of the entire attractor.

If someone is bored and would like to discuss this, feel free to email me.

You can also imagine a similar thing on binary vectors. There two vectors are "orthogonal" if they share no bits that are set to one. So you can encode huge number of concepts using only small number of bits in modestly sized vectors, and most of them will be orthogonal.