[HN Gopher] Elliptic curve 'murmurations' found with AI
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Elliptic curve 'murmurations' found with AI
Author : Brajeshwar
Score : 92 points
Date : 2024-03-05 15:27 UTC (7 hours 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.
| nomel wrote:
| And then there's the Voder (1939):
| https://www.youtube.com/watch?v=TsdOej_nC1M
| 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
| 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.
| 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.
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