  Hopfield networks and Boltzmann machines (BMs) are fundamental energy-based
neural network models. Recent studies on modern Hopfield networks have broaden
the class of energy functions and led to a unified perspective on general
Hopfield networks including an attention module. In this letter, we consider
the BM counterparts of modern Hopfield networks using the associated energy
functions, and study their salient properties from a trainability perspective.
In particular, the energy function corresponding to the attention module
naturally introduces a novel BM, which we refer to as the attentional BM
(AttnBM). We verify that AttnBM has a tractable likelihood function and
gradient for certain special cases and is easy to train. Moreover, we reveal
the hidden connections between AttnBM and some single-layer models, namely the
Gaussian--Bernoulli restricted BM and the denoising autoencoder with softmax
units coming from denoising score matching. We also investigate BMs introduced
by other energy functions and show that the energy function of dense
associative memory models gives BMs belonging to Exponential Family Harmoniums.
