[HN Gopher] Statistics Postdoc Tames Decades-Old Geometry Problem
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Statistics Postdoc Tames Decades-Old Geometry Problem
Author : rbanffy
Score : 159 points
Date : 2021-03-02 13:10 UTC (9 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| tpoacher wrote:
| "To the surprise of experts in the field, a postdoctoral
| statistician has solved ..."
|
| Wth? Am I the only one bothered by this opening statement? A
| post-doc IS an expert in the field. One who has spent a
| significant number of years doing a PhD to become THE expert in
| their own particular subfield, and presumably many more years
| after that diving into even greater detail.
|
| Since when have we grown so accustomed to postdocs that we're
| treating them as on par with undergraduates in terms of academic
| value, and are so super surprised when they make an important
| discovery?
| Fomite wrote:
| As others have mentioned, and as someone who does statistics, I
| think you're putting the emphasis on the wrong word in
| "postdoctoral statistician".
|
| If I solved a major open problem in geometry, Quanta could
| easily write "To the surprise of experts in the field, a
| professor of epidemiology has solved..." and they'd be correct
| in doing so.
| ttymck wrote:
| Is it not referring to the fact that they are a postdoctoral
| _statistician_ and not a "geometrician" or something of that
| nature?
| omaranto wrote:
| Geometer, not geometrician, but yes, you're right.
| strangeloops85 wrote:
| I think you may be misunderstanding the implication. It's not
| that he's a postdoc that's surprising, but that he's a
| statistician who happens to have solved a major problem in
| convex geometry. The following quote sums it up:
|
| "Chen is not a convex geometer by training -- instead, he is a
| statistician who became interested in the KLS conjecture
| because he wanted to get a handle on random sampling. "No one
| knows Yuansi Chen in our community," Eldan said. "It's pretty
| cool that you have this guy coming out of nowhere, solving one
| of [our] most important problems.""
|
| The great thing is, the reaction of the community working on
| these type of problems was to understand and verify the result.
| And once it was, all credit to him.
| tpoacher wrote:
| Thanks for this explanation. That makes it slightly better I
| guess. It still sounds subtly presumptive though, when placed
| as the subheading. It feels a bit like when journalists
| report female scientists and feel the need to report their
| marital status in the first paragraph.
| bopbeepboop wrote:
| It's not like that at all.
|
| It's like reporting "chemist solves problem in particle
| physics" -- which is interesting and a bit unexpected, but
| not at all an unrelated factoid about the person in the way
| marital status would be.
| gnulinux wrote:
| Absolutely not at all. Statistics and geometry are very
| different fields. Sure they both use some part of
| mathematics and an undergrad level statistics and geometry
| curriculum will have many common classes, but at post-doc
| level it is definitely surprising. I'm not saying a
| statistician is not qualified to do research in geometry --
| they might be competent enough to do so -- it's just
| surprising because statistics and geometry work on
| different problems.
| necubi wrote:
| The surprising thing is not that he's a post-doc, it's that
| he's a statistician.
| ska wrote:
| Others have pointed out the misread here, but I'll add that
| typically the thing you are actually expert in after this
| amount of study is typically very narrow. So when they say "the
| surprise of experts in the field" on many topics "the experts"
| they are referring to is often a few dozen people.
| a1369209993 wrote:
| > A post-doc IS an expert in the field.
|
| Other people have already mentioned it, but to phrase it
| another way: he's a expert in a _different_ field.
| OmicronCeti wrote:
| Basically everything quanta writes is so good. They have some of
| the best science writers around in my opinion, and their efforts
| to make science and discovery accessible to non-experts and the
| public is admirable. I highly recommend their podcast that breaks
| down some of the popular stories, as well as "The Joy of X" which
| are long-form interviews with leading scientists.
| strangeloops85 wrote:
| Completely agree! I've really enjoyed Erica Klarreich's writing
| in particular. I think she may very well be one of the most
| gifted and capable writers today on math in particular.
| angry-tempest wrote:
| She interviewed just the right people too! She is like the
| equivalent of Ed Yong for math
| strangeloops85 wrote:
| It helps (and perhaps is no surprise, given her easy
| facility with the material she covers) that she's a Math
| PhD as well :)
| melling wrote:
| Funded by billionaire mathematician and hedge fund founder,
| where the algorithms make all the decisions
|
| https://m.youtube.com/watch?v=QNznD9hMEh0
|
| 1976 winner of the Oswald Veblen Prize in Geometry
|
| https://en.m.wikipedia.org/wiki/Oswald_Veblen_Prize_in_Geome...
| thechao wrote:
| The person referenced above is "James Simon".
| superbcarrot wrote:
| Simons*
|
| Mostly known for being the founder of Renaissance
| Technologies which is one of the largest and most
| successful hedge funds.
| angry-tempest wrote:
| Also the Simons Foundation, which funds the Simons
| Institute, which is just fantastic
| thomasahle wrote:
| They're entertaining to read, but I've never come away from one
| with any sort of understanding of the actual math and how I
| might use it. It would help if they at least defined the
| problem and the new theorem.
| mcguire wrote:
| Did they define " _substantial area_ " and I just missed it?
| btilly wrote:
| You missed it.
|
| _Bourgain guessed that some of these lower-dimensional
| slices must have substantial area. In particular, he
| conjectured that there is some universal constant,
| independent of the dimension, such that every shape
| contains at least one slice with area greater than this
| constant._
|
| In other words there exists C > 0 such that if the
| n-dimensional hypervolume of an n-dimensional convex shape
| is 1, then there must be an n-1'th dimensional slice of
| n-1-dimensional hypervolume at least C.
|
| What was proven is weaker. For any e > 0 there is an N such
| that if N < n, then any n-dimensional convex shape of
| n-dimensional hypervolume 1 must have an n-1'th dimensional
| slice of n-1-dimensional hypervolume at least 1/n^e.
|
| It turns out that the exact things that were proven are
| good enough to improve our bounds on how quickly various
| machine learning algorithms will converge. Which means we
| aren't just hoping based on how they worked in a few
| examples, we have a theory explaining it.
| mbeex wrote:
| No, and in particular by speaking about slices with a lower
| dimension d-1 it becomes not immediately clear, what their
| volume has to do with the one in dimension d of the
| original body, mentioned at the begin of the article.
| btilly wrote:
| They actually did a better job of that with this problem than
| I usually see.
|
| But if you don't have a substantial background, it may be
| hard to track.
| MaxBarraclough wrote:
| I figure this is because they're trying to make it
| accessible. As a somewhat mathematically literate non-
| mathematician I doubt I'd get much out of the real formulae
| that stump today's researchers. Quanta do a good job at
| dumbing it down enough to make sense to someone like me
| without making it completely empty. More specifically, Erica
| Klarreich does a good job of it - she seems to write most if
| not all of the Quanta articles that end up on the HN front
| page.
| revel wrote:
| Aside from being a wonderful achievement and advancement, this is
| a really excellent article. It's very impressive to read such
| technical material presented in such a beautiful way.
| leafmeal wrote:
| Quanta is free, and virtually all of their articles read like
| this.
| zellyn wrote:
| The final paragraph was beautiful...
| paulpauper wrote:
| It seems like unsolved problems in math are being solved at an
| increasingly fast rate. I think a combination of the internet
| making information more readily accessible combined with having
| more people alive to work on such problems, are contributing
| factors to this.
| tovej wrote:
| As far as I can tell, this is a relevant proof when searching
| through high-dimensional parameter spaces (e.g. machine
| learning).
|
| This would mean that overall geometry is not a (big) factor when
| it comes to getting stuck on local optima. Depending on how a
| random walk is implemented however, the conditions might create a
| practically concave search space.
| 6gvONxR4sf7o wrote:
| When it comes to getting stuck at a local optimum, I think it's
| the convexity of the loss function that matters, not just the
| convexity of the parameter space. As I understand it, this
| result says that for convex losses, some simpler samplers work
| near enough to ideally.
| scythmic_waves wrote:
| > This would mean that overall geometry is not a (big) factor
| when it comes to getting stuck on local optima.
|
| I thought the results only applied to convex shapes. Search
| spaces in ML need not be convex, right? Or am I missing
| something?
| tovej wrote:
| I assume they are convex, typically cuboid. This is the type
| of space you get when each parameter is searched for a
| certain range.
|
| Surely there could be search spaces that aren't convex. In
| that case the range of a variable would depend on the values
| of other variables. If you have an example of such a case I'd
| be interested in knowing about it.
| Bootvis wrote:
| If you have linear constraints the search space will be
| convex.
| scythmic_waves wrote:
| Ah I was confusing the loss landscape with the parameter
| search space because you said "local optima". Yep, I
| imagine most parameter search spaces are convex.
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