  It has been proposed by many researchers that combining deep neural networks
with graphical models can create more efficient and better regularized
composite models. The main difficulties in implementing this in practice are
associated with a discrepancy in suitable learning objectives as well as with
the necessity of approximations for the inference. In this work we take one of
the simplest inference methods, a truncated max-product Belief Propagation, and
add what is necessary to make it a proper component of a deep learning model:
We connect it to learning formulations with losses on marginals and compute the
backprop operation. This BP-Layer can be used as the final or an intermediate
block in convolutional neural networks (CNNs), allowing us to design a
hierarchical model composing BP inference and CNNs at different scale levels.
The model is applicable to a range of dense prediction problems, is
well-trainable and provides parameter-efficient and robust solutions in stereo,
optical flow and semantic segmentation.
