This also reaffirms my (wishful) thinking that if there’s a way to do FTL communication it’ll be something with an absurdly tiny factor like 2^-182 with a slight asymmetry in a probability somewhere.
Then you’re not violating FTL, just gaining a very slight chance that you might know something FTL – probably.
Given that c is the speed of causality itself, FTL communications would effectively be like predicting the future.
From that angle, beating light speed by some absurdly tiny factor would probably correspond to a means of predicting the future at some almost absurdly tiny factor better than random guessing.
Edit: Actually...it doesn't make sense to call this FTL communication, it's just predicting the future state of a system given some previous state. FTL comms would have to be predicting the future state of a system without information about the previous state.
Practically speaking predictive modeling would be a means of compensating for light speed comms, kind of like branch prediction in processors or speculative decoding in LLMs, but that wouldn't actually be FTL comms.
Exactly! It’s not forbidden, just very unlikely and would be very strange.
It’d likely involve exponentially more energy as well. It’d be a good sci-if plot point if FTL communications required machines the size of Jupyter to get a few milliseconds of prescience.
You can bootstrap a tiny duration of prescience into arbitrary durations by passing back the same message over and over as many times as you like.
If you know what will happen in one minute, write down the message you see yourself writing down in one minute. In a minute, do the same thing. Now you can pass messages back two minutes.
I mean, being pedantic a little, we don't actually know if c is constant, since measuring c is rather difficult. If c is not in fact constant in some medium or environment, then a huge number of things get very weird very fast.
So, yes, we could've measured c wrong. We just would have no idea if we did.
Source: Veritasium did a very fascinating video explaining this problem.
The relative difference is so absurdly small to be irrelevant at any realisable input size. At least that's my read; e.g. even at n=10^80 (~number atoms in universe), the relative difference is ~0. That's still probably underselling how similar this is to n log n.
this is how all the proofs today are looking, they seems so minimal even when compared to minor improvements in these fields from that last 10 years im wondering why even publish these and not just make research notes public
I love this, entirely separate from any applications or even understanding. It's incredible that we needed this trillion-dollar technology to learn about a faster way to multiply two numbers!
Math is incredibly rich, and even the simplest things have insanely complicated structure when you zoom in. However this all ends up, math is bigger than LLMs, and the people who claim it is getting "solved" and we are running out of open problems haven't stared into the abyss enough.
Sure, all of this might end up being very unpleasant, and I'm glad not to be a mathematician right now. But that's still better than a future where there aren't even any questions left that we can understand and an AI can't solve.
Also, it still seems that AI has a much different style from humans, with more brute force and using obscure literature results, and the future might still end up human/AI complementary. We aren't in an AlphaZero situation where the AI learns everything through self-play. (Yet? But we don't even seem to be moving that way much? Can anybody qualified help out?) Things are just moving really fast now and it's hard to process everything.
What I meant is e.g. the unit distance construction, which (per mathematicial comments) needed a combination of distant fields, so nobody had the necessary expertise. That's different from working within one obscure field.
If you view them as "theories of computational limits" instead of "proposed practical speedups" they can be a lot more interesting.
It's most interesting when the lower bound can actually be proven. In lack of that, we have to guess what the best possible algorithm might yield (generalized or not). This tells us that need not be O(n log n) and we have the opportunity to still find better algorithms than we typically thought would be possible. This does the latter, which is interesting, but it just leaves us to hunger more for what the real limit must be :).
i understand this, but it always feels like we are being tricked when they say "integer multiplication below nlogn" because we intuit that that must mean "faster integer multiplication below nlogn EVERYWHERE!". but in reality it comes with 15 asterisks about the conditions that must be true for their statement to hold true.
Your issue is that I am viewing this proof as what it really is in terms of progressing the field and not from an imaginative perspective. I think that it is important to ground our selves somewhat in reality when discussing research like this because at the end of the day open ai is not doing for fun either.
openai wants to show the world what their product can do and i am simply not impressed
I can respect your opinion if it's consistent- i.e. it's not just directed at OpenAI's results. But this seems to be an example of a common phenomenon with AI discourse: while disparaging LLM achievements, you indirectly insult the careers/accomplishments of 99% of mathematicians for whom this would easily be the crown jewel of their CV.
This progresses the field a great deal, just perhaps not the field you're interested in? There's nothing wrong with a "and what can I apply that to in my life tomorrow" approach but it's certainly not the only approach worth having in the world.
O(n lg n) is a bit of a threshold value. For a lot of algorithms, this is the best you can do, even in theory (similar to how O(n^2) is also a threshold for many algorithms). So for many algorithms, people stop trying when they get close to O(n lg n) on the belief that you'll never do better than that.
The fact that you can in principle go faster than n lg n, even if just by an almost imperceptible amount, is kind of surprising. It raises the question of, if n lg n isn't the limit, what is? How far down can we get the speed? If we can get it a little past n lg n, maybe we can go a lot further.
[or at least that is my understanding. not a theoretical computer scientist]
It's like when Tony Hawk did a 900 for the first time. Now 900s in skateboarding aren't a big deal, kids can do it now. It was proving to the world what was possible was the mental hurdle that inspires others to actually try at the problem harder.
Is there an associated machine-checked proof of this?
We're in full vibe-code mode at work, so I understand both how powerful frontier models can be and how often they can over-confidently state subtly (or not so subtly) wrong things, even when you're taking great efforts to try to keep that from happening.
So without a Lean development or extensive human verification, I guess I'm a little bit skeptical, and even sort of hoping this is wrong - not just because of my not so positive feelings about AI, but by my disposition towards beauty in math. n log n is an awful lot nicer than what we have here.
Agree 100% on wanting machinr verification of AI generated math.
But in regards to beauty, i feel like multiplication already has a lot of non beautiful exponents. Best known matrix multiply is O(n^2.371). For integer factorization, the inverse of this problem, general number field sieve is a crazy subexponential.
If factorization is just barely subexponential, is it really that surprising that multiplication is just barely sub n lg n ?
I mean, cracking anything below the nlogn bound implies that there might be much more room for improvement. Often a very minor win over the theory opens up enough extra attention to later truly move the needle.
It's 50 pages and cites this other paper in the same repo:
OpenAI. An explicit power saving for the exact discrete Fourier transform.
Here's a random excerpt:
8.3 The middle transform and the final permutation
The factor QFt in (35) can be computed from a cyclic convolution and two pointwise phase multiplications. The chirp identity below performs the frequency change in Q without applying Q as a separate permutation of the array. The second identity shows how the retained source permutation R cancels when computing a convolution. Here ∗ denotes cyclic convolution on the product of the coordinate groups and a dot denotes coordinatewise multiplication.
This also reaffirms my (wishful) thinking that if there’s a way to do FTL communication it’ll be something with an absurdly tiny factor like 2^-182 with a slight asymmetry in a probability somewhere.
Then you’re not violating FTL, just gaining a very slight chance that you might know something FTL – probably.
From that angle, beating light speed by some absurdly tiny factor would probably correspond to a means of predicting the future at some almost absurdly tiny factor better than random guessing.
Edit: Actually...it doesn't make sense to call this FTL communication, it's just predicting the future state of a system given some previous state. FTL comms would have to be predicting the future state of a system without information about the previous state.
Practically speaking predictive modeling would be a means of compensating for light speed comms, kind of like branch prediction in processors or speculative decoding in LLMs, but that wouldn't actually be FTL comms.
It’d likely involve exponentially more energy as well. It’d be a good sci-if plot point if FTL communications required machines the size of Jupyter to get a few milliseconds of prescience.
If you know what will happen in one minute, write down the message you see yourself writing down in one minute. In a minute, do the same thing. Now you can pass messages back two minutes.
So, yes, we could've measured c wrong. We just would have no idea if we did.
Source: Veritasium did a very fascinating video explaining this problem.
Math is incredibly rich, and even the simplest things have insanely complicated structure when you zoom in. However this all ends up, math is bigger than LLMs, and the people who claim it is getting "solved" and we are running out of open problems haven't stared into the abyss enough.
Also, it still seems that AI has a much different style from humans, with more brute force and using obscure literature results, and the future might still end up human/AI complementary. We aren't in an AlphaZero situation where the AI learns everything through self-play. (Yet? But we don't even seem to be moving that way much? Can anybody qualified help out?) Things are just moving really fast now and it's hard to process everything.
Is that true? Look at the average math paper on arxiv. Of course it's obscure to those not in the exact sub-field - math has become very specialized.
i will NEVER care about proposed multiplication speedups unless they are truly generalized
It's most interesting when the lower bound can actually be proven. In lack of that, we have to guess what the best possible algorithm might yield (generalized or not). This tells us that need not be O(n log n) and we have the opportunity to still find better algorithms than we typically thought would be possible. This does the latter, which is interesting, but it just leaves us to hunger more for what the real limit must be :).
Your issue is that I am viewing this proof as what it really is in terms of progressing the field and not from an imaginative perspective. I think that it is important to ground our selves somewhat in reality when discussing research like this because at the end of the day open ai is not doing for fun either.
openai wants to show the world what their product can do and i am simply not impressed
The fact that you can in principle go faster than n lg n, even if just by an almost imperceptible amount, is kind of surprising. It raises the question of, if n lg n isn't the limit, what is? How far down can we get the speed? If we can get it a little past n lg n, maybe we can go a lot further.
[or at least that is my understanding. not a theoretical computer scientist]
We're in full vibe-code mode at work, so I understand both how powerful frontier models can be and how often they can over-confidently state subtly (or not so subtly) wrong things, even when you're taking great efforts to try to keep that from happening.
So without a Lean development or extensive human verification, I guess I'm a little bit skeptical, and even sort of hoping this is wrong - not just because of my not so positive feelings about AI, but by my disposition towards beauty in math. n log n is an awful lot nicer than what we have here.
But in regards to beauty, i feel like multiplication already has a lot of non beautiful exponents. Best known matrix multiply is O(n^2.371). For integer factorization, the inverse of this problem, general number field sieve is a crazy subexponential.
If factorization is just barely subexponential, is it really that surprising that multiplication is just barely sub n lg n ?
1. https://github.com/openai/math/blob/main/preprints/Matrix-Mu...
Sharing AI Progress in Mathematics
https://news.ycombinator.com/item?id=49984923
OpenAI. An explicit power saving for the exact discrete Fourier transform.
Here's a random excerpt:
8.3 The middle transform and the final permutation The factor QFt in (35) can be computed from a cyclic convolution and two pointwise phase multiplications. The chirp identity below performs the frequency change in Q without applying Q as a separate permutation of the array. The second identity shows how the retained source permutation R cancels when computing a convolution. Here ∗ denotes cyclic convolution on the product of the coordinate groups and a dot denotes coordinatewise multiplication.