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Sampling point uniformly at random satisfying equality constraints

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 … 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 …
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