Displacement convexity — A useful framework for the study of spatially coupled codes

Spatial coupling has recently emerged as a powerful paradigm to construct graphical models that work well under low-complexity message-passing algorithms. Although much progress has been made on the analysis of spatially coupled models under message passing, there is still room for improvement, both in terms of simplifying existing proofs as well as in terms of proving additional properties. We introduce one further tool for the analysis, namely the concept of displacement convexity. This concept plays a crucial role in the theory of optimal transport and it is also well suited for the analysis of spatially coupled systems. In cases where the concept applies, displacement convexity allows functionals of distributions which are not convex to be represented in an alternative form, so that they are convex with respect to the new parametrization. The alternative convex structure can then often be used to prove the uniqueness of the minimizer of this functional. As a proof of concept we consider spatially coupled (l, r)-regular Gallager ensembles when transmission takes place over the binary erasure channel. In particular, we first show the existence of an optimal profile which minimizes the potential functional governing this system. This profile characterizes the “decoding wave” of the spatially coupled system. We then show that the potential function of the coupled system is displacement convex. Due to some translational degrees of freedom the convexity by itself falls short of establishing the uniqueness of the minimizing profile. But as we will discuss it is an important step in this direction.