Euclidean network information theory

Abstract

Many network information theory problems face the similar difficulty of single letterization. We argue that this is due to the lack of a geometric structure on the space of probability distributions. In this thesis, we develop such a structure by assuming that the distributions of interest are all close to each other. Under this assumption, the Kullback-Leibler (K-L) divergence is reduced to the squared Euclidean metric in an Euclidean space. In addition, we construct the notion of coordinate and inner product, which will facilitate solving communication problems. We will present the application of this approach to the point-to-point channels, general broadcast channels (BC), multiple access channels (MAC) with common sources, interference channels, and multi-hop layered communication networks without or with feedback. It can be shown that with this approach, information theory problems, such as the single-letterization, can be reduced to some linear algebra problems. Solving these linear algebra problems, we will show that for the general broadcast channels, transmitting the common message to receivers can be formulated as the trade-off between linear systems. We also provide an example to visualize this trade-off in a geometric way. For the MAC with common sources, we observe a coherent combining gain due to the cooperation between transmitters, and this gain can be obtained quantitively by applying our technique. In addition, the developments of the broadcast channels and multiple access channels suggest a trade-off relation between generating common messages for multiple users and transmitting them as the common sources to exploit the coherent combining gain, when optimizing the throughputs of communication networks. To study the structure of this trade-off and understand its role in optimizing the network throughput, we construct a deterministic model by our local approach that captures the critical channel parameters and well models the network. With this deterministic model, for multi-hop layered networks, we analyze the optimal network throughputs, and illustrate what kinds of common messages should be generated to achieve the optimal throughputs. Our results provide the insight of how users in a network should cooperate with each other to transmit information efficiently.