3 ms·

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

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.

MaxBarraclough 6y ago
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.
mcguire 6y ago
Did they define "substantial area" and I just missed it?
- mbeex 6y ago
  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 6y ago
  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 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.
btilly 6y ago
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 6y ago

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.

mcguire 6y ago

Did they define "substantial area" and I just missed it?

mbeex 6y ago
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 6y ago
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 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.

btilly 6y ago

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.