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Given a real rectangular matrix $X$, I would like to uniformly sample from the set of real rectangular matrices $\mathbb{M}$ that satisfy $||X-S||\leq \epsilon, \forall S\in\mathbb{M}$ and for a fixed real, scalar radius $\epsilon$.

Now procedurally/algorithmically, how can I obtain a matrix $\hat{S}$ (sample) by such a 'uniform sampling' within the specified radius? The key is that the sampling should be uniform when obtaining an instance $\hat{S}$. What would be the necessary steps to be taken to be able to do this?

The fact that the set $\mathbb{M}$ seems very large is why the notion of sampling really confuses me to be able to assign a probability law although the $\epsilon$ does constraint the set to an extent.

The following is what I am trying to do, i.e get a sample of the type $X_k$ as shown in the screenshots attached below: enter image description here enter image description here

halms
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  • Maybe you can do a uniform sample in X and discard those that not respect your restriction. – Rcoster Apr 22 '13 at 12:04
  • I think you should make precise what you mean by "uniform sampling", as I can think of a few possible interpretations. – Nate Eldredge Apr 22 '13 at 12:21
  • what norm are you using ? – mike Apr 23 '13 at 12:52
  • @mike Euclidean Norm – halms Apr 23 '13 at 15:17
  • @halms: That can't be what you mean. There are infinitely many such matrices; if they are all equally likely, that just says each one has probability zero. – Nate Eldredge Apr 23 '13 at 15:32
  • @NateEldredge I meant that $\forall S \in \mathbb{M}$ satisfying the $\epsilon$ constraint, each matrix has a equal probability of being sampled in the sampling procedure. I see you having a deep probabilistic background, leading to different view points, but this is what i meant. Also for the case with column matrices as well, I would need a procedure/algorithm. Now, after you put forth this question, I thought a bit and was wondering about a special cases as well: i) The case when the set $\mathbb{M}$ has more structural properties like lying on the Steifel or Grassman manifolds – halms Apr 23 '13 at 15:32
  • @NateEldredge I attached a picture in the question, talking about a sample of this sort. I think this will help you understand it better, in case I might have missed something although I agree with what you are saying. Please look at this screenshots in terms of the sample $X_k$ as mentioned over there. There is also a set of points of differentiability $D$ of a chosen function $f(.)$ that is also considered. – halms Apr 23 '13 at 16:07
  • this is no different from drawing a point uniformly from the unit ball in $\mathbb R^{m = n^2}$. Get one uniform on the sphere by sampling i.i.d. normals and normalizing to be 1. Then multiply by something that gives right radius. – mike Apr 23 '13 at 18:24
  • @mike can you post it as an answer? Also, were you suggesting it with the view point of a column vector or for matrices, in terms of the normalization of an i.i.d normal sample followed by multiplying with the radius? – halms Apr 23 '13 at 19:00
  • I think if you are using euclidean norm it doesn't make any difference, the matrices are the same as $\mathbf R^{N^2}$. – mike Apr 23 '13 at 20:19
  • @Mike I tried your answer and after the multiplication with the radius, the sphere is centered around zero, I guess the sphere has to also be translated to the required center which would be the vectorized matrix X – halms Apr 23 '13 at 23:31
  • @NateEldredge Your answer over here http://math.stackexchange.com/questions/87230/picking-random-points-in-the-volume-of-sphere-with-uniform-probability helps me out! – halms Apr 24 '13 at 22:18

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You can draw uniformly a matrix on the space $Q = \{ S: \forall (i,j), \vert S_{i,j} - X_{i,j} \vert \leq \epsilon \} \subset \mathbb M$ and then keep only the matrices in $\mathbb M$. Although, I have the feeling that the probability of the event $ (S \in \mathbb M) \vert (S \in Q)$ is very small when the size of the $X$ is big...

roger
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  • This is like Rejection sampling, and you are right, the expected number of trials required would grow to be very large/intractable – halms Apr 23 '13 at 20:41