空间耦合标量递归的位移凸性
Displacement Convexity in Spatially Coupled Scalar Recursions
摘要：
我们介绍了一般的空间耦合系统，由标量递归的分析技术。这样的系统可以用一个势泛函的变分形式来表示。我们表明，在温和的条件下，潜在的功能是\emph{displacement convex}和极小给出的递归的不动点。此外，我们给出了系统的条件，使最小化不动点是唯一的，直到沿空间方向的平移。该条件与\cite{KRU12}中关于空间不动点存在的条件相匹配。\emph{Displacement convexity}适用于编码理论、压缩感知、随机约束满足问题以及统计力学模型中出现的各种空间耦合递归。我们说明它与应用程序的低密度奇偶校验和广义LDPC码用于传输的二进制擦除信道，或一般的二进制无记忆对称信道内的高斯倒易信道近似，以及压缩感知。
英文摘要：
We introduce a technique for the analysis of general spatially coupled systems that are governed by scalar recursions. Such systems can be expressed in variational form in terms of a potential functional. We show, under mild conditions, that the potential functional is \emph{displacement convex} and that the minimizers are given by the fixed points of the recursions. Furthermore, we give the conditions on the system such that the minimizing fixed point is unique up to translation along the spatial direction. The condition matches those in \cite{KRU12} for the existence of spatial fixed points. \emph{Displacement convexity} applies to a wide range of spatially coupled recursions appearing in coding theory, compressive sensing, random constraint satisfaction problems, as well as statistical mechanical models. We illustrate it with applications to Low-Density Parity-Check and generalized LDPC codes used for transmission on the binary erasure channel, or general binary memoryless symmetric channels within the Gaussian reciprocal channel approximation, as well as compressive sensing.
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