A Dichotomy of Functions in Distributed Coding: An Information Spectral Approach

TL;DR

This paper investigates distributed data compression for function computation with sources that have memory and may not be stationary, establishing conditions under which the rate region matches the classical Slepian-Wolf region, and extending existing results.

Contribution

It introduces a new class of sources called smooth sources and provides a comprehensive dichotomy theorem for distributed function computation, generalizing prior work to sources with memory.

Findings

Achievable rate region matches Slepian-Wolf region for smooth sources.

Necessary and sufficient conditions for symbol-wise functions.

Analysis of error probability in moderate deviation regime.

Abstract

The problem of distributed data compression for function computation is considered, where (i) the function to be computed is not necessarily symbol-wise function and (ii) the information source has memory and may not be stationary nor ergodic. We introduce the class of smooth sources and give a sufficient condition on functions so that the achievable rate region for computing coincides with the Slepian-Wolf region (i.e., the rate region for reproducing the entire source) for any smooth sources. Moreover, for symbol-wise functions, the necessary and sufficient condition for the coincidence is established. Our result for the full side-information case is a generalization of the result by Ahlswede and Csiszar to sources with memory; our dichotomy theorem is different from Han and Kobayashi's dichotomy theorem, which reveals an effect of memory in distributed function computation. All results are given not only for fixed-length coding but also for variable-length coding in a unified manner. Furthermore, for the full side-information case, the error probability in the moderate deviation regime is also investigated.