How to sample subspaces uniformly

  • Thread starter Thread starter wdlang
  • Start date Start date
  • Tags Tags
    Subspaces
AI Thread Summary
To sample N-dimensional subspaces of an M-dimensional linear space uniformly, one proposed method involves generating an M*N matrix with elements sampled from a normal distribution. After creating this matrix, QR decomposition can be applied to obtain N orthonormal vectors that span the desired subspace. There is also a mention of using the Haar measure on the space of unitary matrices, which provides a natural probability measure for selecting subspaces. The discussion highlights the need for clarity on whether a computationally feasible uniform sampling method or a more natural distribution is preferred. Overall, the QR decomposition approach is suggested as a potential numerical method for achieving uniform sampling of subspaces.
wdlang
Messages
306
Reaction score
0
i need to sample the N-dimensional subspaces of a M-dimensional linear space over C uniformly.

That is, all subspaces are sampled with equal probability

how should i do it?

would this work? First generate a M*N matrix, the real and imaginary parts of each element is sampled from the normal distribution. Then by using QR decomposition, i can get N orthonormal vectors, which span a N-dimensional subspace. Take it.
 
Physics news on Phys.org
Are you looking for something reasonably "uniform" that you can put into a computer, or are you looking for a distribution that feels somehow natural?

If the former, I have no idea, and other people may have better ideas.

If the latter, there must be a natural choice, given a fixed basis for the big space. Say we're looking at a random N-dimensional subspace of \mathbb C^M (the latter with its standard basis). Let \mathcal U(M) denote the space of unitary M\times M matrices. There's a very natural probability measure on \mathcal U(M), called the Haar measure \mu. I would think of a "uniformly chosen N-dimensional subspace" as picking a random U\in \mathcal U(M) via \mu and then picking the span of the first N columns of U.

*[It has the property that for any collection \mathcal S \subseteq \mathcal U(M) of matrices with defined probability, and any matrix U\in \mathcal U(M), we have \mu(U\mathcal S)= \mu(\mathcal S). Moreover, \mu is unique in satisfying this property.]
 
  • Like
Likes 1 person
economicsnerd said:
Are you looking for something reasonably "uniform" that you can put into a computer, or are you looking for a distribution that feels somehow natural?

If the former, I have no idea, and other people may have better ideas.

If the latter, there must be a natural choice, given a fixed basis for the big space. Say we're looking at a random N-dimensional subspace of \mathbb C^M (the latter with its standard basis). Let \mathcal U(M) denote the space of unitary M\times M matrices. There's a very natural probability measure on \mathcal U(M), called the Haar measure \mu. I would think of a "uniformly chosen N-dimensional subspace" as picking a random U\in \mathcal U(M) via \mu and then picking the span of the first N columns of U.

*[It has the property that for any collection \mathcal S \subseteq \mathcal U(M) of matrices with defined probability, and any matrix U\in \mathcal U(M), we have \mu(U\mathcal S)= \mu(\mathcal S). Moreover, \mu is unique in satisfying this property.]

i am actually looking for a numerical method.

my idea is the following. First generate a M*N matrix, the real and imaginary parts of each element of which are drawn independently from the normal distribution. Then do QR decomposition to get the N orthonormal basis vectors spanning the subspace.

i am not sure whether this is the right algorithm.
 

Similar threads

Replies
13
Views
3K
Replies
5
Views
2K
Replies
5
Views
3K
Replies
7
Views
2K
Replies
4
Views
2K
Back
Top