基于一般旋转不变编码矩阵的VAMP译码下的稀疏叠加码
Sparse superposition codes under VAMP decoding with generic rotational invariant coding matrices
摘要：
稀疏叠加码最初是作为一种在高斯信道上实现容量的通信方案而提出的，其编码矩阵由独立同分布组成。我们将该编码方案扩展到具有任意频谱的旋转不变编码矩阵的更通用的集合，其包括高斯集合作为特殊情况。我们进一步介绍和分析了一种基于向量近似消息传递（VAMP）的译码器，基于标准的副本对称势理论和状态演化分析，我们的主要发现是某些结构化的编码矩阵集合（如部分行正交）与i.i.d.矩阵，以及VAMP阈值的频谱独立上限。最重要的是，我们推导出一个简单的“频谱标准”的计划是在同一时间的容量实现，同时具有最好的算法阈值，在“大部分大小”的渐近极限。因此，我们的研究结果提供了实用的设计原则，在这个有前途的通信方案的编码矩阵。
英文摘要：
Sparse superposition codes were originally proposed as a capacity-achieving communication scheme over the gaussian channel, whose coding matrices were made of i.i.d. gaussian entries.We extend this coding scheme to more generic ensembles of rotational invariant coding matrices with arbitrary spectrum, which include the gaussian ensemble as a special case. We further introduce and analyse a decoder based on vector approximate message-passing (VAMP).Our main findings, based on both a standard replica symmetric potential theory and state evolution analysis, are the superiority of certain structured ensembles of coding matrices (such as partial row-orthogonal) when compared to i.i.d. matrices, as well as a spectrum-independent upper bound on VAMP's threshold. Most importantly, we derive a simple "spectral criterion " for the scheme to be at the same time capacity-achieving while having the best possible algorithmic threshold, in the "large section size" asymptotic limit. Our results therefore provide practical design principles for the coding matrices in this promising communication scheme.
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