Computation Over Gaussian Networks With Orthogonal Components

TL;DR

This paper develops coding strategies for computing functions like sums and histograms over Gaussian networks with orthogonal components, characterizing their computation capacities.

Contribution

It introduces a novel approach combining lattice codes and linear network coding to compute functions over Gaussian networks and characterizes their capacities.

Findings

Computation capacity is exactly characterized for orthogonal Gaussian networks.

Approximate capacity is derived for networks with multiple-access components.

The method applies to a broad class of functions including sums and histograms.

Abstract

Function computation of arbitrarily correlated discrete sources over Gaussian networks with orthogonal components is studied. Two classes of functions are considered: the arithmetic sum function and the type function. The arithmetic sum function in this paper is defined as a set of multiple weighted arithmetic sums, which includes averaging of the sources and estimating each of the sources as special cases. The type or frequency histogram function counts the number of occurrences of each argument, which yields many important statistics such as mean, variance, maximum, minimum, median, and so on. The proposed computation coding first abstracts Gaussian networks into the corresponding modulo sum multiple-access channels via nested lattice codes and linear network coding and then computes the desired function by using linear Slepian-Wolf source coding. For orthogonal Gaussian networks (with no broadcast and multiple-access components), the computation capacity is characterized for a class of networks. For Gaussian networks with multiple-access components (but no broadcast), an approximate computation capacity is characterized for a class of networks.