Statistics Postdoc Tames Decades-Old Geometry Problem
quantamagazine.org
quantamagazine.org
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...
Mostly known for being the founder of Renaissance Technologies which is one of the largest and most successful hedge funds.
(Albert Schwarz formulated Chern–Simons theory, a topological quantum field theory, using the Chern-Simons form)
But if you don't have a substantial background, it may be hard to track.
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 ε > 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^ε.
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.
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?
"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.
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.
To use your example, it's more like saying "experts surprised female chemist solves problem in particle physics", when "chemist solves problem in particle physics" would have been so much better.
But I take your point.
“Postdoctoral”?
I’m genuinely confused here.
It sounds like “experts surprised professional chemist solves problem in particle physics”.
“Professional chemist” is a meaningful title, unlike “female chemist”. The problem with naked “chemist” is that there’s a lot of grades — from student to hobbyist to YouTuber to postdoc to PI for industry lab.
How are you interpreting “postdoc” the same way as “female”?
One if those is a job; one of those is a biological fact.
Other people have already mentioned it, but to phrase it another way: he's a expert in a different field.
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.
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.
I thought the results only applied to convex shapes. Search spaces in ML need not be convex, right? Or am I missing something?
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.