For a matrix $X\in\mathbb{R}^{d\times n}$, what condition can I impose on $X$ to make the collection of its columns generic in the sense that they look like the result of uniformly sampling a convex region in $\mathbb{R}^d$? Ideas or relevant papers are welcome and appreciated. Thanks.

  • $\begingroup$ So, the question is on uniformly sampling a convex region, right? $\endgroup$ Oct 26, 2019 at 5:55
  • $\begingroup$ No, the question is on "what conditions I can impose on X" to make its columns look like uniform point cloud on a convex region. $\endgroup$
    – Min Wu
    Oct 26, 2019 at 5:58
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    $\begingroup$ Why make it look like the original if you can get the original? $\endgroup$ Oct 26, 2019 at 6:28
  • $\begingroup$ You want something like a likelihood measure, that given $X$ will tell you how "likely" it is to come from a uniform sampling on a convex? As it is, it is unclear what you are asking. Any distribution of points may come from such a sampling, although possibly with a low likelihood. $\endgroup$ Oct 26, 2019 at 8:23
  • $\begingroup$ Sorry for the confusions. I will restate the problem again. $\endgroup$
    – Min Wu
    Oct 26, 2019 at 16:46


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