Monotone deep Boltzmann machines

TL;DR

This paper introduces monotone deep Boltzmann machines, a new class of probabilistic models that enable efficient approximate inference with fully-general weight structures, expanding the capabilities of deep generative models.

Contribution

The paper proposes the monotone DBM, allowing arbitrary self-connections and restricted weights to ensure unique mean-field fixed points, enabling efficient inference in fully-general DBMs.

Findings

Allows for efficient approximate inference in fully-general weight DBMs.

Enables joint image completion and classification within a single probabilistic model.

Avoids mean-field inference pitfalls in traditional RBMs.

Abstract

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.