Just came across this discussion. I'm one of the authors, and if anyone is interested, I'd be happy to answer any questions about the paper (jason@math.stanford.edu). It originated in me hearing from Andrew Ng and other folks in the ML community about how powerful deep belief nets were, and wanting to quantify that power more explicitly. The first step, for mathematicians, is to understand what a RBM can represent. As you point out, the paper is written for a math journal and so emphasizes certain things. I presented a poster from a more ML perspective at the ICML workshops. Cheers!