Maybe it isn't, but I have found the literature on the topic to be very clear in general. I do not find section 1.5 and 2.3 explicit at all.

Section 2.3 reads like a redefinition of the exponential family and its link to the maximum entropy principle. I think the idea is to optimize the sufficient statistics, but it's not clear.

Section 1.5... well you talk about two dimensional translational symmetries, and then make a link with position and momentum, but where is that coming from? This seems merely like an artifact of looking at two dimensional translational symmetries, and not at the general case. Again, what does the quantum analogy bring to the table? I think - though it's not clear - that what you're suggesting is drawing the latent noise following a distribution which is reflects the expected symmetry in the manifold.

So what is the takeout? Are you attempting to parametrize max-entropy distribution using the symmetries as constraints?

Not me dude but I like underdogs and outsiders. What u saying sounds right: symmetries, like the ones computed in section 3.8 are used as constraints a.k.a. "quantum numbers" describing the states in the latent layer. That is vintage quantum mechanics. The part is theoretical and not in the theano code by looks of it, so harder to comment on... The quantum analogy also brings along the laplacian form of the conditional density in section 1.5, so there u go. On your comment, which "literature on the topic" is clear? Have u read the original VAE paper http://arxiv.org/pdf/1401.4082.pdf. 95% of paper and most of 21 equations are about gradients, and gaussian too, they even invent a new term: stochastic back-propagation, why, who needs that? All the math is done in the Gibbs/ACE paper in 4 lines, just compute your cost bound and back-propagate? Having seen this earlier would have saved me a lot of pondering. Pythagorean theorem does sound too simple, I give u that...