[HN Gopher] Elliptic curve 'murmurations' found with AI
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Elliptic curve 'murmurations' found with AI
Author : Brajeshwar
Score : 286 points
Date : 2024-03-05 15:27 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| factormeta wrote:
| Seems coincide with this that was on HN:
| http://www.incompleteideas.net/IncIdeas/BitterLesson.html
| galkk wrote:
| I love story in spite of the article above.
|
| Speech synthesis also was attempted as modeling of human
| biology: computer modeling of throats, vocal cords, how the air
| is going through mouth.
|
| In the end computational power also won. No need of all of
| that.
| acer4666 wrote:
| The article is talking about deep learning winning, ie neural
| networks. Surely modelling of human biology is part of that?
| nomel wrote:
| Maybe emergent, to some extent, but not explicit.
| bigyikes wrote:
| Neural nets do not model human biology, they are models
| which are inspired by human biology.
| nomel wrote:
| And then there's the Voder (1939):
| https://www.youtube.com/watch?v=TsdOej_nC1M
| totetsu wrote:
| Maybe we didn't need to, but I'm glad someone did
| https://m.youtube.com/watch?v=qobhDJ_vEOc
| bbor wrote:
| > The biggest lesson that can be read from 70 years of AI
| research is that general methods that leverage computation are
| ultimately the most effective, and by a large margin.
|
| It's nice when an author includes a sentence up top that
| betrays their standpoint so that I can stop reading. I'm sure
| this person is very nice and has lots of stuff to say, but this
| is the same old Scruffy v. Neat fight, except now the former
| side thinks that they're empirically completely right. Which
| doesn't even make sense, they're not mutually exclusive claims,
| and to say that the result of 70 years of expert systems is any
| kind of failure is just revisionist.
|
| For the same reason, I don't read many papers about Realism vs
| Idealism, Nature vs Nurture, etc
| SonOfLilit wrote:
| I recommend that you continue. It's a very short and highly
| influential piece by a guy at the top of his field, you'll
| take a thing or two from the one-minute read even if you
| agree with nothing he says.
| bbor wrote:
| Why thanks for the suggestion, that's very kind. I just
| read it - in fact I think I read this early on in my
| research on scruffies v neats.
|
| My takeaway now is the same, sadly. It's not so much that I
| disagree with his premises that I find his whole attitude
| and conclusion to be a preposterous artifact of ego
| inflation after helping found a line of research that was
| much more productive than people thought it would be. I get
| it, that's very exciting, but I need _way_ more evidence
| than that to completely abandon self-conscious structured
| reasoning in my conception of a good AGI, much less the
| human mind. Like this: This is a big
| lesson. As a field, we still have not thoroughly learned
| it, as we are continuing to make the same kind of mistakes.
|
| This is just arrogance. You don't see this among
| philosophers or social scientists, who recognize that
| rhetoric is more than the cherry on top of science, and
| that this sort of confidence is dangerous. To see
| "designing things by hand" as a categorical "mistake" is
| just... that's a hot take.
|
| But either way we're all on the same side. Despite my harsh
| words I'm glad he helped pave the way LLMs, which are the
| biggest unexpected breakthrough in our lifetimes IMO. Which
| understandably makes people confident
| couchand wrote:
| This is a great story that highlights how human beings working
| together can reveal new insights. I love how the author covers
| each individual's contribution to the discovery.
|
| It's also interesting to see how critical the human element of
| this story is, and how incidental the "AI" piece is. A computer
| system employed statistics to exploit (but not comprehend) a
| pattern in a high-dimensional dataset. This led researchers to
| examine the relevant dimensions using traditional data
| visualization tools.
|
| Once the nature of the pattern was characterized, other
| mathematicians were able to use their insight to find deep
| connections to other areas. These interconnections are now
| blossoming.
| bbor wrote:
| The title is obviously clickbait, but the idea a good one I
| hope some of the scientists here take away from this: "using
| AI" is about identifying things it can do that you could never
| hope to, usually for reasons of scale or complexity. LLM-based
| systems will revolutionize the day-to-day of science IMO, but
| that doesn't mean that they're replacing human reasoning
| faculties.
| naasking wrote:
| > A computer system employed statistics to exploit (but not
| comprehend) a pattern in a high-dimensional dataset.
|
| Nitpick: I don't like this phrasing because there are degrees
| of comprehension, and understanding that two or more things are
| correlated in specific ways is a form of comprehension.
| "Exploit but not explain" is a better phrasing IMO.
| robertk wrote:
| Very cool result but the title is overselling the "AI"
| contribution. It seems like they trained a few standard binary
| classifiers (Naive Bayes, decision trees, kNN). The novelty is
| the independent variable coming from an attribute precomputed for
| many known elliptic curves in the LMFDB database, namely the
| Dirichlet coefficients of the associated L-function; and the
| dependent variable being whether or not the elliptic curve has
| complex multiplication (CM), an important theoretical property
| for which lots of flashy theorems begin with assuming whether or
| not the curve has CM. They go on to train another binary
| classifier (and a separate size k classifier) to determine a
| curve's Sato-Tate identity component using the Euler coefficients
| and group-theoretic information about the Sato-Tate group
| (constructed by randomly sampling elements and representing the
| two non-trivial coefficients of their characteristic polynomials
| as independent variables in the classifier). They also run a PCA:
| https://arxiv.org/pdf/2010.01213.pdf
|
| The cool part is that they then stepped back and scratched their
| heads wondering why the classifier was so good at achieving
| separation for these dependent variables in the first place, and
| plotting the points showed them to be (non-linearly) separable
| due to a visually clear pattern! The punchline and the reason
| it's so important to understand these data points, the Euler
| coefficients for elliptic curves, is because they contain all the
| relevant number-theoretic information about the curve. With some
| major handwaving, understanding them perfectly would lead to
| things like the Langlands program (and some analogues of the
| Riemann hypothesis) getting resolved. These wide reaching
| conjectures are ultimately structural assertions about
| L-functions, and L-functions are uniquely specified by their
| Euler coefficients (the a_p term in their Euler factors). Will
| murmurations help with that? Who knows, but the more patterns the
| better for forming precise conjectures.
|
| Relevant intersectional credentials: I have lead ML engineering
| teams in industry and also did my doctorate work in this area of
| math, including using the LMFDB database referenced in the
| article for my research (which was much smaller back then and has
| grown a lot, so very neat to see it's still a force for empirical
| findings!).
| frakt0x90 wrote:
| This is something I've been thinking about a lot lately.
| Especially in combinatorics and number theory, there are
| databases like oeis, LMFDB, etc that contain tons of data with
| the ability to generate more algorithmically (sometimes easier
| said than done). Using ML to get heuristics and really good
| guesses on where the next opportunities lie and then
| formalizing it once you have a good guess would be SO cool.
|
| Is there a name for that? Or groups working on that stuff that
| I could follow?
|
| My own little pet project was I scraped OEIS and built a graph
| of sequences where 2 were connected if one mentioned the other
| in its related sequences section. You got these huge clusters
| around prime powers and other important sequences. Then I
| thought maybe you could use a GNN to do link prediction
| providing an estimation of a relationship that should exist but
| hasn't been discovered yet.
| jononor wrote:
| In these area of physics informed machine learning this is
| refered to as "discovering new physics". Probably there are
| analogs in computational mathematics, biology, chemistry,
| etc.
| goodmachine wrote:
| IANAM but I guess the name for mining OEIS or generating
| scads of data iteratively for analysis would be empirical
| mathematics.
|
| It's empirical metamathematics if you attempt this with
| networks of axioms/theories
|
| https://www.wolframscience.com/metamathematics/empirical-
| met...
|
| https://writings.stephenwolfram.com/2020/09/the-empirical-
| me...
| joachimma wrote:
| I am not a mathematician but have some interest on a pop-sci
| level. I believe this presentation at G-Research by Alex
| Davies would be of interest.
| https://www.youtube.com/watch?v=Mp_skPK-X9M
| ykonstant wrote:
| The Lean 4 Focused Research Organization has ML
| interoperability in its roadmap. Since Lean 4 is shaping up
| to be a capable general purpose language as well, I can
| imagine a Lean project that retrieves and formats LMFDB data,
| uses it to train and test a NN, gets Lean 4 proof code from
| it, verifies or rejects it (possibly with more detailed
| feedback) and loops this like a "conversation".
|
| However, Lean 4 still has a long way to go in terms of speed
| and library features, and I at least have given up on writing
| optimized code until we get the new compiler (whose timeline
| seems optimistic to me, but Leo de Moura knows much better).
| knotthebest wrote:
| At which point would mathematicians become obsolete?
| Something like this seems like it could automate a lot of
| mathematics research, no?
| djbusby wrote:
| Suppose someone understands 0% of that. What would I type into
| DDG or Wikipedia to start?
|
| Like, ecliptic curves are part of libsoduim/nacl - does it mean
| something "big"?
| couchand wrote:
| As someone who understands about 2% of the GP but maybe 85%
| of TFA, I'd suggest diving into the various topics explored
| there. Galois Fields, for instance, are a rich topic for
| Wikipedia research and have intuitive and surprising
| properties that make them fun to learn about.
|
| This will lead you deeper into study of abstract algebra
| concepts like groups and rings. If you haven't done much set
| theory you will probably go deep on that and develop an
| opinion on the Axiom of Choice.
|
| Then you'll probably surface a bit to look at elliptic curves
| and consider their many applications in abstract and concrete
| topics like cryptography and the elusive proof of Fermat's
| Last Theorem.
|
| By then you'll have caught up to me. In the meantime I'll be
| reading up on module forms and L-functions.
| tanvach wrote:
| I highly recommend the PeakMath
| (https://youtube.com/@PeakMathLandscape?si=zQg6bbp2SvfqzKYm)
| RH saga video series on YouTube for this topic.
|
| They are excellent, and not requiring more than high school
| maths knowledge to really get quite deep into the mysterious
| connections between prime numbers, Riemann hypothesis,
| elliptic curves and L-Functions.
| ykonstant wrote:
| I second this recommendation; it is serious material made
| very accessible. The channel is great, and this series is
| truly a marvel.
|
| However, while it does not require more _knowledge_ than
| high school math, it does require more _maturity_ and
| certainly lots of patience.
| brabel wrote:
| > Very cool result but the title is overselling the "AI"
| contribution. It seems like they trained a few standard binary
| classifiers (Naive Bayes, decision trees, kNN).
|
| But it seems they would never have even suspected there were
| such patterns if the "AI" had not provided evidence for them?
|
| By the way: the tools mentioned, like decision trees, Bayes and
| kNN were all taught in the AI course I attended one and a half
| decade ago... AI was basically ML at the time, but nowadays it
| seems that ML has become "just statistics", and AI only
| includes LLMs.
| radicalbyte wrote:
| There are plenty of companies using ML methods (DT, Bayes,
| kNN), normal NN etc now that the AI money spigot is wide
| open, if only as part of the "shit in, shit out" process.
| weebull wrote:
| Sounds like it's far more about "big" data analysis, and
| recognising that elyptic curves encryption has a statistically
| apparent signature. AI/ML was just the analysis that exposed
| it.
| billiam wrote:
| The article is actually a great illustration of how far ahead of
| machine learning humans remain in their ability to collaborate
| and make intuitive connections (the AI contribution was minimal).
| Which LLM is going to say, hey this pattern looks like the birds
| out my window, or the problem I worked on years ago and never got
| anywhere, or I must send a competing LLM a preprint of my paper?
| naasking wrote:
| > Which LLM is going to say, hey this pattern looks like the
| birds out my window, or the problem I worked on years ago and
| never got anywhere, or I must send a competing LLM a preprint
| of my paper?
|
| They might never say that, but the model has a good chance of
| containing that association because learning is compression. If
| two things have the same patterns, they will likely be tied to
| the same networks in the model, because that's just how good
| compression works.
| notfed wrote:
| "found with AI" is the new "made in Rust"?
| weebull wrote:
| "Made in rust." is dying off?
| carlossouza wrote:
| > Sutherland was impressed by the significant dose of luck that
| had led to the discovery of murmurations.
|
| Talented people + hard work... + LUCK!
|
| > Even then, the murmurations were only found because of
| Pozdnyakov's inexperience.
|
| Also, fresh inexperienced eyes to see what experts would dismiss!
|
| What a great read :)
| lwansbrough wrote:
| So might this be a precursor to cracking ECC?
|
| Assume they go on to find a formula which defines the
| relationship between a_p and rank, what does that actually
| achieve?
| omidHeravi wrote:
| Next thing you know, it's solved the discrete log problem and
| the rest of the millennium problems.
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