How can we randomly generate matrix $A \in \mathbb{R}^{n \times m}_{\geq 0}$ that satisfies
$A 1_n = m1$ and $A^T 1_m = n1$.
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3$\begingroup$ You should specify the distribution of the randomly generated matrices. I think that you mean equidistributed in some sense. $\endgroup$Dieter Kadelka– Dieter Kadelka2022-02-08 12:43:22 +00:00Commented Feb 8, 2022 at 12:43
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2$\begingroup$ This is generalization to the question mathoverflow.net/questions/73805/…, which is open. $\endgroup$Amritanshu Prasad– Amritanshu Prasad2022-02-09 06:28:34 +00:00Commented Feb 9, 2022 at 6:28
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2 Answers
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I don't know if this is what you want, but here are two differnet ways:
In the case $m=n$: Generate a bunch of random permutation matrices $A_i$ and take a random linear combination $\sum_i \alpha_i A_i$ with $\sum \alpha_i=n$ and $\alpha_i\geq 0$.
Generate a random $m\times n$ matrix and use the Sinkhorn scaling algorithm the equalize row and column sums to the desired values.
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$\begingroup$ With $\alpha_i\ge0$. $\endgroup$Gerry Myerson– Gerry Myerson2022-02-08 22:46:27 +00:00Commented Feb 8, 2022 at 22:46
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$\begingroup$ This is not going to be uniform. $\endgroup$Amritanshu Prasad– Amritanshu Prasad2022-02-09 06:30:27 +00:00Commented Feb 9, 2022 at 6:30
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$\begingroup$ Thanks @GerryMyerson! $\endgroup$Dirk– Dirk2022-02-09 08:48:33 +00:00Commented Feb 9, 2022 at 8:48
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$\begingroup$ @AmritanshuPrasad Sure - but there was no indication of any distribution given and my constructions are for sure random (I have no idea what kind of distribution then give…). $\endgroup$Dirk– Dirk2022-02-09 08:49:34 +00:00Commented Feb 9, 2022 at 8:49
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$\begingroup$ The other answer by Brenday McKay is quite relevant and related to my first point: You want to sample from a generalized Birkhoff polytope: en.wikipedia.org/wiki/Doubly_stochastic_matrix#Generalisations $\endgroup$Dirk– Dirk2022-02-09 08:51:39 +00:00Commented Feb 9, 2022 at 8:51
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This is an example of generating a random point in a polytope. There is some work on that general problem, but I don't know if it would be practical for your special case. For example:
Rubin, Generating random points in a polytope, Communications in Statistics - Simulation and Computation, 13 (1984) 375-396.
Probably searching forward and backwards from that will uncover other relevant articles.
