Toward a Fundamental Theory of Gaussian Source-Channel Networks

过去60年的研究使人们对点对点通信问题有了深刻的理解。然而，现代通信系统的体系结构通常涉及网络，而不是各个信源到各个目的地的孤立的直接连接。这项研究计划的长期目标是对复杂的通信网络进行类似的全面理解。作为实现这一目标的重要一步，本研究旨在提供一个理论框架来描述高斯源-信道网络(即，由具有二次代价约束的矢量高斯信道和具有二次失真度量的矢量高斯源组成的网络模型)的基本性能极限以及相关编码方案的结构。这一理论框架包括三个主要部分：归约技巧、极值不等式和非线性方法。具体地，利用归约技术将给定的高斯源-信道网络转换为具有更易于分析的统计结构的等价模型，然后利用极值不等式对等价模型的性能极限进行有界，最后利用非线性方法证明所得界的紧性。提出的研究将通过引入新的分析技术和解决重要的开放问题来丰富网络信息理论的基础。它还将促进对复杂网络系统的理解，建议具体的编码方案和广泛的体系结构原则。约简技术的发展有可能在信息论和理论计算机科学之间架起一座桥梁。通过这个项目得到的极值不等式具有独立的数学意义，并且很可能在信息论之外找到应用。在教育方面，本项目将为快速发展的网络信息理论领域的研究生提供一个有价值的研究和学习环境，并为本科生的课程开发和培养做出贡献。

Research over the past six decades has led to a deep understanding of the point-to-point communication problem. However, the architecture of modern communication systems typically involves networks, rather than isolated direct connections of individual sources to individual destinations. The long-term goal of this research program is to develop a similarly comprehensive understanding of complex communication networks. As an important step toward this goal, the proposed research is intended to provide a theoretical framework for characterizing fundamental performance limits of Gaussian source-channel networks (i.e., network models that consist of vector Gaussian channels with quadratic cost constraint and vector Gaussian sources with quadratic distortion measure) and the structure of associated coding schemes. This theoretical framework is envisaged to contain three major components: reduction techniques, extremal inequalities, and nonlinear methods. Specifically, reduction techniques are used to convert the given Gaussian source-channel network to an equivalent model with a statistical structure that is more amenable to analysis, then the performance limit of the equivalent model is bounded with the aid of extremal inequalities, and finally nonlinear methods are invoked to prove the tightness of the resulting bound. The proposed research will enrich the fundamentals of network information theory through the introduction of new analysis techniques and the solution of important open problems. It will also advance the understanding of complex networked systems, suggesting both specific coding schemes and broad architectural principles. The development of reduction techniques has the potential to provide a bridge between information theory and theoretical computer science. Extremal inequalities obtained through this project are of independent mathematical interest and are likely to find applications beyond information theory. On the education front, this project will provide a valuable research and learning environment for graduate students in the rapidly progressing field of network information theory, and contribute to course development and training of undergraduate students.