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 Post subject: random projectionsPosted: Tue, 26 Jun 2012 15:32:30 UTC
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Joined: Thu, 7 Jun 2012 11:13:33 UTC
Posts: 4
Let P be a random orthogonal projection on R^d, chosen according to the "uniform distribution" on the set of rank k orthogonal projections. Are the squares of any two distinct entries on the diagonal uncorrelated?

(It probably doesn't matter but I assume that k is at least 2 and d at least 2k.)

Any ideas?

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 Post subject: Re: random projectionsPosted: Fri, 29 Jun 2012 03:52:20 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Lucky Hans wrote:
Let P be a random orthogonal projection on R^d, chosen according to the "uniform distribution" on the set of rank k orthogonal projections. Are the squares of any two distinct entries on the diagonal uncorrelated?

(It probably doesn't matter but I assume that k is at least 2 and d at least 2k.)

Any ideas?

Obviously .

Let's do a simple calculation with d=4 and k=2, which has a reasonably nice coordinate system to work with.

The Haar measure on SO(4) is of course induced by . So upon identifying , we might as well take 1,i as the two orthonormal vectors, and have where are random unit quaternions and is the projection to . Now, writing , , we get

Switching to (spherical) polars, (and for brevity , , etc.)

(, )
So

So now take expectations of , and remember the (normalised) measure on is given by and you should see the covariance is nonzero. (You can cheat using Weyl's integration formula instead of passing to the explicit description of Spin(4).)

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 Post subject: Re: random projectionsPosted: Fri, 29 Jun 2012 20:39:50 UTC
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Joined: Thu, 7 Jun 2012 11:13:33 UTC
Posts: 4
ok, thanks.

I have another adjacent question. I want to estimate E|p1-t p2| from below, where p1 is the squared norm of first column and p2 of the second column, respectively, and t between 0 and one. Taking the modulus hurts me. I need an estimate from below of the form c/d, where c is a positive constant that is independent on the ambient dimension d. (It may depend on the rank k of the orthogonal projections.) I am sure that it is correct. I am pretty sure that the term dE|p1-t p2| grows for growing d and fixed k..

Any ideas?

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 Post subject: Re: random projectionsPosted: Sat, 30 Jun 2012 10:01:19 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
Whoa, hold on dude. If you've been hurt by a modulus, consult your doctor first.

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