I was a little informal with my argument. It's not strictly true that we only see C = 0.2 when K = 2. I was reading what the graph says about the case when N is much greater than k. I'll try to clarify.
C is meant to be the smallest constant so that, for each (N, k, epsilon) with k > C epsilon^-2 log N and epsilon > 0, there exists some arrangement of N vectors in R^k with inner products smaller than epsilon in absolute value. For each (N, k), the author optimized epsilon and reported the empirical value k epsilon^2 / log N, which is the smallest value of C for which our condition holds restricted to the given values of (N, k). (Of course, this is my good-faith interpretation---the article introduces C in the context of a JL distortion bound, and it takes an extra step to turn that into a bound on inner products.)
It can be shown that C = 4 satisfies this condition, when log is the natural log. See [1], for example. Based on the graph, the article claims to do better: "for sufficiently large spaces," it says we can put C = 0.2. This would be a very significant improvement.
For k = 2, the graph shows that C will be lower than 0.2 for sufficiently large N. (The color scheme is confusing! The line for k = 2 is the one that starts at just under 0.8 when N = 1.) Already for k = 3, the graph doesn't give us reason to believe it will be lower than 0.2---you correctly observed it gets to around 0.3. For larger value of k, the graph doesn't seem to show what we can expect for large N: the curves go up, but do not come down. This is what I meant with my comment: the conclusion that C <= 0.2 as N -> infinity is only justified by the behavior at K = 2.
Now, do these results make sense? In the case k = 2, we're trying to put a large number (N) of vectors on the unit circle, and thinking about how small the maximum inner product (epsilon) between any pair of vectors can be. As N -> infinity, the vectors will be packed very tightly and the maximum inner product epsilon will come close to 1. Overall, C = k epsilon^2 / log N will become arbitrarily small. In fact, the same happens for every k.
So, just in connection to this graph, the article makes three mistakes:
1) The article's interpretation of its experiment is wrong: the graph alone doesn't show that C < 0.2 for "large spaces".
2) However, it should be obvious a priori that, for all values of k, the reported values C should converge to 0 for large N (albeit very slowly, at a rate of 1/log N).
3) Unfortunately, this doesn't tell us anything about the minimum value of k / log(N) for a given epsilon and k, and so it doesn't support the conclusion of the article.
The problem with this kind of LLM-driven article is that it gives uncareful work the _appearance_ of careful work but none of the other qualities that usually come with care.
There's a lot of beautiful writing on these topics on the "pure math" side, but it's hard to figure out what results are important for deep learning and to put them in a form that doesn't take too much of an investment in pure math.
I think the first chapter of [1] is a good introduction to general facts about high-dimensional stuff. I think this is where I first learned about "high-dimensional oranges" and so on.
For something more specifically about the problem of "packing data into a vector" in the context of deep learning, last year I wrote a blog post meant to give some exposition [2].
One really nice approach to this general subject is to think in terms of information theory. For example, take the fact that, for a fixed epsilon > 0, we can find exp(C d) vectors in R^d with all pairwise inner products smaller than epsilon in absolute value. (Here C is some constant depending on epsilon.) People usually find this surprising geometrically. But now, say you want to communicate a symbol by transmitting d numbers through a Gaussian channel. Information theory says that, on average, I should be able to use these d numbers to transmit C d nats of information. (C is called the channel capacity, and depends on the magnitude of the noise and e.g. the range of values I can transmit.) The statement that there exist exp(C d) vectors with small inner products is related to a certain simple protocol to transmit a symbol from an alphabet of size exp(C d) with small error rate. (I'm being quite informal with the constants C.)
> The implications of these geometric properties are staggering. Let's consider a simple way to estimate how many quasi-orthogonal vectors can fit in a k-dimensional space. If we define F as the degrees of freedom from orthogonality (90° - desired angle), we can approximate the number of vectors as [...]
If you're just looking at minimum angles between vectors, you're doing spherical codes. So this article is an analysis of spherical codes… that doesn't reference any work on spherical codes… seems to be written in large part by a language model… and has a bunch of basic inconsistencies that make me doubt its conclusions. For example: in the graph showing the values of C for different values of K and N, is the x axis K or N? The caption says the x axis is N, the number of vectors, but later they say the value C = 0.2 was found for "very large spaces," and in the graph we only get C = 0.2 when N = 30,000 and K = 2---that is, 30,000 vectors in two dimensions! On the other hand, if the x axis is K, then this article is extrapolating a measurement done for 2 vectors in 30,000 dimensions to the case of 10^200 vectors in 12,888 dimensions, which obviously is absurd.
I want to stay positive and friendly about people's work, but the amount of LLM-driven stuff on HN is getting really overwhelming.
> Our architectural choices are closely aligned with principles observed in biological brains.
How? They point out three design choices: linear attention, MoE layers, and spike coding.
Apparently linear attention is brain-inspired because it can be viewed as a "simplified abstraction of dendritic dynamics with multi-branch morphology." Who knows what that means exactly [1]. They don't discuss it further. MoE layers apparently reflect "a principle of modular specialization." Fine, whatever.
Now, using a dozen attention variants + MoE is bog standard. The real novelty would be spike coding. Page 11 is dedicated to the different ways they could turn signals into spike trains, including such biologically-inspired mechanisms as using two's complement. However, they don't actually do spike coding in a time domain. In their implementation, "spike coding" apparently means to turn activations into integers. Section 3.3.3 claims that this lets us simulate an underlying spiking neural network, so we can validate the spiking approach without using special hardware. But if your SNN can be simulated faithfully on a GPU by turning things into integers, isn't that a bit of a depressing SNN?
Either I'm missing something, or this is just just dressing standard techniques with loads of meaningless jargon. Of course that’s a very popular way to operate in deep learning nowadays.
[1] Like, attention can draw from multiple tokens, sort of like how different spines of a dendrite can draw from multiple axons? Can’t make this stuff up.
One way to understand why without writing down the CDF/PDF:
When X is an exponential variable and c is a constant, X + c has the same distribution as X after conditioning on large outcomes. In other words, these two variables have same "tail." This is true exactly for exponential distributions. (Sometimes this is called "memorylessness.")
Similarly, when U has a uniform distribution on [0, 1] and c is a constant, cU has the same distribution as U after conditioning on small outcomes.
But if cU is distributed like U near 0, then -ln(c U) is distributed like -ln(U) near infinity. But -ln(c U) = -ln(c) - ln(U), so the tail of -ln(U) doesn't change when we add a constant, meaning it must have an exponential distribution.