These weekly brief notes are gems of current issues, events, writers, and information, with myriad links to sources and resources, all on top of the Crown jewels of your own shared knowledge about our economic history and the Utopian possibilities within it. Thank you.
Well I am the first to admit that the properties of high-dimensional spaces surprise me. But I don't think you have got the Johnson–Lindenstrauss lemma right. At least if I'm reading you correctly.
It is a fact that has nothing to do with Johnson–Lindenstrauss that two random vectors u and v drawn independently from very high-dimensional normal or uniform distributions have, with high probability, a) almost the same length and b) are almost orthogonal. What Johnson–Lindenstrauss says is that if you choose a random orthogonal projection of u and v from the original d-dimensional space down to a much lower d'-dimensional space, then with high probability the length of u - v will be close to the length of u' - v'. "Close" and "high probability" are sharply defined concepts related to d' in such a way that if you fix one then you can trade off the other two.
This highly convenient fact is useful in applications well outside AI; if it weren't true the sort of high-dimensional Monte Carlo I work on would be hopeless. But there is an excellent discussion of this in the context of AI that comes all the way from ... UC Berkeley: https://people.eecs.berkeley.edu/~jrs/189/lec/22.pdf. You could probably hunt down the author and talk to them directly!
>>a way of separating over-optimistic investors from their wealth—rather than FILTH—”Failed in London, Try Hong Kong”—we have FICTA—”Failed in Crypto, Try AI”<<
LOL- Thanks! I very much enjoy the AI insights expressed here... ..the discussion of market valuations in general... like graph w/Tesla & market top 10 P/E
Love it!! The NVIDIA and Tesla comments are perfect.....
These weekly brief notes are gems of current issues, events, writers, and information, with myriad links to sources and resources, all on top of the Crown jewels of your own shared knowledge about our economic history and the Utopian possibilities within it. Thank you.
Well I am the first to admit that the properties of high-dimensional spaces surprise me. But I don't think you have got the Johnson–Lindenstrauss lemma right. At least if I'm reading you correctly.
It is a fact that has nothing to do with Johnson–Lindenstrauss that two random vectors u and v drawn independently from very high-dimensional normal or uniform distributions have, with high probability, a) almost the same length and b) are almost orthogonal. What Johnson–Lindenstrauss says is that if you choose a random orthogonal projection of u and v from the original d-dimensional space down to a much lower d'-dimensional space, then with high probability the length of u - v will be close to the length of u' - v'. "Close" and "high probability" are sharply defined concepts related to d' in such a way that if you fix one then you can trade off the other two.
This highly convenient fact is useful in applications well outside AI; if it weren't true the sort of high-dimensional Monte Carlo I work on would be hopeless. But there is an excellent discussion of this in the context of AI that comes all the way from ... UC Berkeley: https://people.eecs.berkeley.edu/~jrs/189/lec/22.pdf. You could probably hunt down the author and talk to them directly!
>>a way of separating over-optimistic investors from their wealth—rather than FILTH—”Failed in London, Try Hong Kong”—we have FICTA—”Failed in Crypto, Try AI”<<
LOL- Thanks! I very much enjoy the AI insights expressed here... ..the discussion of market valuations in general... like graph w/Tesla & market top 10 P/E
Goats ... maybe ask Ada Palmer?