In this context, the Hamiltonian is just the log of the conditional probability of the input data given some parameters. In fact, such functions are much more d

In this context, the Hamiltonian is just the log of the conditional probability of the input data given some parameters. In fact, such functions are much more diverse than the functions defined by neural networks (which, though there are many variants, are basically all compositions of linear and nonlinear functions). The question being asked in the paper is which Hamiltonians can be robustly approximated by neural networks. The authors then argue that the class of Hamiltonians that appear in nature are simple enough that neural nets do a decent job.