First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem.
The description of my problem is the following:
Sample a point $x$ uniformly at random from the solutions satisfying linear equalities $Ax = b$.
Specifically, I'm interested in sampling a $n$ x $m$ joint probability distribution of two discrete random variables that satisfies given marginal probabilities. This is a special case of the more general described above.
Also, this problem is similar in sampling contingency tables with fixed margins, but it is not restricted to integral solutions.
What I have found / tried so far:
Uniformly Sampling from Convex Polytopes (not exactly what I want, as I'm only interested about on the boundaries and not in the polytope)
Random Sampling a linearly constrained region in n-dimensions... (This is closer to what I want, but can (?) only be applied to the case of $n$ x $2$ dimensional probability distributions)
http://www.mzlabs.com/JMPubs/Sampling%20Contingency%20Tables-Dyer.pdf (Paper about sampling contingency tables with fixed margins)
http://www.mathworks.com/matlabcentral/fileexchange/34208-uniform-distribution-over-a-convex-polytope/content/cprnd.m (Implementation of a method to sample a point that satisfies a set of linear inequalities. I tried enforcing equality constraints but the method does not work)