  Deep Boltzmann machines (DBMs), one of the first ``deep'' learning methods
ever studied, are multi-layered probabilistic models governed by a pairwise
energy function that describes the likelihood of all variables/nodes in the
network. In practice, DBMs are often constrained, i.e., via the
\emph{restricted} Boltzmann machine (RBM) architecture (which does not permit
intra-layer connections), in order to allow for more efficient inference. In
this work, we revisit the generic DBM approach, and ask the question: are there
other possible restrictions to their design that would enable efficient
(approximate) inference? In particular, we develop a new class of restricted
model, the monotone DBM, which allows for arbitrary self-connection in each
layer, but restricts the \emph{weights} in a manner that guarantees the
existence and global uniqueness of a mean-field fixed point. To do this, we
leverage tools from the recently-proposed monotone Deep Equilibrium model and
show that a particular choice of activation results in a fixed-point iteration
that gives a variational mean-field solution. While this approach is still
largely conceptual, it is the first architecture that allows for efficient
approximate inference in fully-general weight structures for DBMs. We apply
this approach to simple deep convolutional Boltzmann architectures and
demonstrate that it allows for tasks such as the joint completion and
classification of images, within a single deep probabilistic setting, while
avoiding the pitfalls of mean-field inference in traditional RBMs.
