A Structured Coding Framework for Communication and Computation Over Continuous Networks

This work considers an information-theoretic characterization of the set of achievable rates, costs, and distortions in a broad class of distributed communication and function computation scenarios with general continuous-valued sources and channels. A framework is presented which involves fine discretization of the source and channel variables followed by communication over the resulting discretized network. In order to evaluate the resulting achievable regions, convergence results for information measures are provided under the proposed discretization process. Prior works have considered such convergence results for mutual information quantities written in terms of univariate functions of random variables, and sums of independent random variables. These convergence results have been used to derive achievable regions in point-to-point communication scenarios and specific multiterminal scenarios with continuous alphabets. However, the best-known achievability results for distributed communication and function computation scenarios, which are based on structured coding strategies, involve mutual information quantities written in terms of bivariate functions of random variables, e.g., sum of two (not necessarily independent) random variables. A main contribution of this work is to show the convergence of mutual information quantities written in terms of sums of quantized random variables. This is an essential step in evaluating the achievable regions in continuous distributed computation scenarios by generalizing the structured coding strategies which have been previously used to derive the best-known achievable regions in discrete networks. The framework is used to provide achievability results for the problems of function computation over multiple-access channels, distributed source coding, function reconstruction (two-help-one), and multiple-descriptions source coding. In each scenario, discrete structured coding strategies along with the aforementioned convergence results are used to derive inner bounds to set of achievable rates, costs, and distortions. Furthermore, structured coding strategies are considered for distributed function computation scenarios involving computation of non-additive functions. The techniques are used to study an example where the objective is to compute the product of channel inputs over a multiple access channel, and an inner bound to the achievable rate region is evaluated. It is shown that, in contrast to many well-studied scenarios in multiterminal information theory, Gaussian input distribution is outperformed by the uniform input distribution.