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.
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.
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.
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.
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.
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.
"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?
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.
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.
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.
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.
Like I said, it's not the news itself, it's the phrasing of the subheading that bothers me.
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.
What word in the subheading do you believe is like “female” in your example?
“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.
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.
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.
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.
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.
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.
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.
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.
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.
