most recent 30 from http://mathoverflow.net 2013-05-22T03:34:07Z http://mathoverflow.net/feeds/user/30475 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118313/approximating-a-subspace-by-sampling-a-base-without-replacement/118331#118331 vxf 2013-01-08T04:15:43Z 2013-01-08T04:15:43Z

<p>I don't understand some things here. First, you did not say that the given matrix was random. So it would seem that you either want an upper or lower bound of some kind. You get wildly different answers depending on how far from mutually orthogonal the columns are. For example, if all the columns are the same, then the distance is 0. Second, you say you want a measure of the distance between two spaces but your metric measures the distance between points and a space. If you want the distance between spaces, I would expect some sort of angle (or angles).</p>

http://mathoverflow.net/questions/118313/approximating-a-subspace-by-sampling-a-base-without-replacement/118331#118331 vxf 2013-01-08T05:09:39Z 2013-01-08T05:09:39Z

So now I think I get the bunbury comment. If we assume that we start with a fixed subspace S and then choose a random sequence of new vectors from the space X (in other words, the matrix is going to be generated at random) and ask for the distribution of angles between these new vectors and S then this is equivalent to choosing a random sequence of unit vectors in X. To answer that, we only have to compute the distribution for a single vector in X. Isn't this the same as putting a hyperplane through the origin of a sphere?

http://mathoverflow.net/questions/118313/approximating-a-subspace-by-sampling-a-base-without-replacement/118331#118331 vxf 2013-01-08T04:52:36Z 2013-01-08T04:52:36Z

I see now that you can not mean to measure the distance between the whole subspace and the sample subspace because that distance is going to be zero as a previous comment has mentioned. Instead, you are looking at the distance between each vector and the whole subspace. However, this depends heavily on the matrix as I mentioned. If the columns in the matrix are orthogonal, then the angles between the vectors and the subspace are either 0 or pi/2.