Comments on: Kernels of random sign matrices http://tcsmath.wordpress.com/2008/05/08/kernels-of-random-sign-matrices/ some mathematics of theoretical computer science Fri, 16 Oct 2009 11:37:51 +0000 http://wordpress.com/ hourly 1 By: tasos http://tcsmath.wordpress.com/2008/05/08/kernels-of-random-sign-matrices/#comment-122 tasos Tue, 14 Apr 2009 18:46:40 +0000 http://tcsmath.wordpress.com/?p=44#comment-122 James, you forgot a sqrt(N) in the statement of theorem 3. James, you forgot a sqrt(N) in the statement of theorem 3.

]]> By: jrluw http://tcsmath.wordpress.com/2008/05/08/kernels-of-random-sign-matrices/#comment-26 jrluw Thu, 12 Jun 2008 18:15:34 +0000 http://tcsmath.wordpress.com/?p=44#comment-26 Boaz, thanks; you are right that the line dist(x’, ker(A)) >= dist(x,ker(A)) - |A||x-x’| doesn't make any sense, and also that the proof doesn't use it. I have changed the description of steps (1)-(3) to reflect what actually goes on in the proof. About Milman's "conjecture" -- it was made informally at a talk he gave last year; I believe he intended to guess that there is no completely explicit construction, but it's not clear what that means. E.g. he would not be happy with an arbitrary deterministic polynomial time algorithm. Note that no one knows how to compute $latex \Delta(X)$ efficiently. In fact, no one knows whether $latex \Delta(X) \geq C$? is in NP, or even how to give witnesses for $latex \Delta(X) \approx \sqrt{N}$. Or even how to generate random $latex X$'s that have a witness with high probability. Boaz, thanks; you are right that the line

dist(x’, ker(A)) >= dist(x,ker(A)) – |A||x-x’|

doesn’t make any sense, and also that the proof doesn’t use it. I have changed the description of steps (1)-(3) to reflect what actually goes on in the proof.

About Milman’s “conjecture” — it was made informally at a talk he gave last year; I believe he intended to guess that there is no completely explicit construction, but it’s not clear what that means. E.g. he would not be happy with an arbitrary deterministic polynomial time algorithm. Note that no one knows how to compute \Delta(X) efficiently. In fact, no one knows whether \Delta(X) \geq C? is in NP, or even how to give witnesses for \Delta(X) \approx \sqrt{N}. Or even how to generate random X’s that have a witness with high probability.

]]> By: Boaz Barak http://tcsmath.wordpress.com/2008/05/08/kernels-of-random-sign-matrices/#comment-25 Boaz Barak Thu, 12 Jun 2008 13:07:07 +0000 http://tcsmath.wordpress.com/?p=44#comment-25 Hi James, Just came across this collection - it's really nice. I hope you do keep up posting these. This is the kind of blog that's actually worth reading. I'm not sure I follow dist(x', ker(A)) >= dist(x,ker(A)) - |A||x-x'| but it doesn't matter much. It seems that the equation you need (and use) is that |x-x'| >= |A(x-x')|/|A| BTW I saw in your (great) powerpoint slides that you mentioned something that Milman conjectures is impossible. I hope you expand on that, and whether he conjectures an information theoretic lower bound, or that it's hard to come up with an explicit construciton. Boaz Hi James,

Just came across this collection – it’s really nice. I hope you do keep up posting these. This is the kind of blog that’s actually worth reading.

I’m not sure I follow
dist(x’, ker(A)) >= dist(x,ker(A)) – |A||x-x’|
but it doesn’t matter much.

It seems that the equation you need (and use) is that
|x-x’| >= |A(x-x’)|/|A|

BTW I saw in your (great) powerpoint slides that you mentioned something that Milman conjectures is impossible. I hope you expand on that, and whether he conjectures an information theoretic lower bound, or that it’s hard to come up with an explicit construciton.

Boaz

]]> By: aravind srinivasan http://tcsmath.wordpress.com/2008/05/08/kernels-of-random-sign-matrices/#comment-14 aravind srinivasan Tue, 13 May 2008 17:56:31 +0000 http://tcsmath.wordpress.com/?p=44#comment-14 just a minor comment -- in Lemma 2, all you need is that EZ >= 0, not that Z is non-negative necessarily; you can apply the Chebyshev-Cantellin inequality then (which will also lead to a slightly smaller denominator, but maybe Paley-Zygmund also leads to it). just a minor comment — in Lemma 2, all you need is that EZ >= 0, not that Z is non-negative necessarily; you can apply the Chebyshev-Cantellin inequality then (which will also lead to a slightly smaller denominator, but maybe Paley-Zygmund also leads to it).

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