Maximum Entropy Functions: Approximate Gacs-Korner for Distributed Compression

TL;DR

This paper introduces an approximate Gács-Körner method for distributed source compression, leveraging spectral graph theory to balance helper rate and source rate reduction in network settings.

Contribution

It develops an efficient approximate Gács-Körner optimization framework for distributed compression, incorporating helper nodes and spectral graph techniques.

Findings

Proposes an efficient spectral algorithm for approximate Gács-Körner optimization.

Establishes a connection between the optimization and maximal correlation coefficient.

Demonstrates trade-offs between helper rate and source compression efficiency.

Abstract

Consider two correlated sources $X$ and $Y$ generated from a joint distribution $p_{X,Y}$. Their G\'acs-K\"orner Common Information, a measure of common information that exploits the combinatorial structure of the distribution $p_{X,Y}$, leads to a source decomposition that exhibits the latent common parts in $X$ and $Y$. Using this source decomposition we construct an efficient distributed compression scheme, which can be efficiently used in the network setting as well. Then, we relax the combinatorial conditions on the source distribution, which results in an efficient scheme with a helper node, which can be thought of as a front-end cache. This relaxation leads to an inherent trade-off between the rate of the helper and the rate reduction at the sources, which we capture by a notion of optimal decomposition. We formulate this as an approximate G\'acs-K\"orner optimization. We then discuss properties of this optimization, and provide connections with the maximal correlation coefficient, as well as an efficient algorithm, both through the application of spectral graph theory to the induced bipartite graph of $p_{X,Y}$.