Comments on: A conversation about complexity lower bounds, X http://gowers.wordpress.com/2009/11/10/a-conversation-about-complexity-lower-bounds-x-2/ Mathematics related discussions Wed, 15 May 2013 08:08:31 +0000 hourly 1 http://wordpress.com/ By: Troy Lee http://gowers.wordpress.com/2009/11/10/a-conversation-about-complexity-lower-bounds-x-2/#comment-4298 Tue, 10 Nov 2009 17:30:17 +0000 http://gowers.wordpress.com/?p=1195#comment-4298 There is a closely related norm which I find it easier to think
about, the \gamma_2 norm, also known as the Schur product operator
norm. This is defined as \gamma_2(A)=\max_{u,v} \|A \circ uv^t\|_{tr}
where the max is over unit vectors, and you take the trace norm of A entrywise
product with the rank one matrix uv^t. From this expression you can
see that the \gamma_2 norm of the 2^k-by-2^k Walsh
matrix is at least 2^{k/2}, taking u,v to be uniform unit
vectors. In fact, this is the exact value.

An equivalent formulation is \gamma_2(A)=min_{XY^t=A} r(X)r(Y)
where r(X) is the largest \ell_2 norm of a row of X.
From this formulation you can see that \gamma_2 of the identity
matrix is at most one (taking X,Y to be identities). Again this is
the exact value.

Using both of these formulations, one can show
\gamma_2(A \otimes B)=\gamma_2(A) \gamma_2(B), which can give
the norm for the intermediate V_{j,k} matrices you talk about.

It is somewhat disheartening that the obvious bound here, that
\gamma_2 can increase by at most one with each row operation, seems
to just give a lower bound of \sqrt{N} operations to transform the
N-by-N Walsh matrix to identity.

]]> By: Kristal Cantwell http://gowers.wordpress.com/2009/11/10/a-conversation-about-complexity-lower-bounds-x-2/#comment-4297 Tue, 10 Nov 2009 16:56:33 +0000 http://gowers.wordpress.com/?p=1195#comment-4297 Here are 54 attempts to resolve P vs NP:

http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

]]> By: gowers http://gowers.wordpress.com/2009/11/10/a-conversation-about-complexity-lower-bounds-x-2/#comment-4295 Tue, 10 Nov 2009 15:06:39 +0000 http://gowers.wordpress.com/?p=1195#comment-4295 The problem of trying to find circuit lower bounds for pretty much anything is pretty much out in my view. I just don’t have any feeling that I had with DHJ(3) that there was an approach that deserved to be investigated. In a sense you could say that these dialogues were a sort of pseudo-Polymath that went nowhere very much.

Having said that, I don’t completely rule out the idea of having a complexity-related Polymath project. There were some interesting questions about Boolean polynomials a few posts back, and one or two questions I quite like in this most recent instalment about trying to formulate more precisely the difference between an NP property that trivially reformulates the problem and one that doesn’t. (The motivation for that would be to have a more general natural-proofs barrier that rules out “potentially fruitful” NP properties and just leaves you with the useless ones.) I also had one or two thoughts about how one might go about trying to find a new public-key cryptosystem, though I was slightly thrown when Richard Lipton recently described this as a mathematical disease. Anyhow, I might ignore that, think a little bit more about my idea, and if I can get it to stop looking hopeless then I might try to turn it into a proposal.

Another idea might be if any experts in theoretical computer science had a good idea for a complexity-type project that is hard enough to be interesting but not super-hard in the way that many of the notorious problems are. (There were one or two suggestions early on — it might be a good idea to discuss them a little further.)

]]> By: Jason Dyer http://gowers.wordpress.com/2009/11/10/a-conversation-about-complexity-lower-bounds-x-2/#comment-4294 Tue, 10 Nov 2009 14:10:27 +0000 http://gowers.wordpress.com/?p=1195#comment-4294 I just wanted to thank you for a wonderful series.

(Also, I find it interesting that this last post is much less technical — at least to me — than many of the other ones.)

Is this problem still a potential polymath, or is it out?

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