Conditional graph entropy as an alternating minimization problem

TL;DR

This paper reformulates conditional graph entropy as an alternating minimization problem, enabling a simple iterative algorithm for its computation and revealing its connection to a broader convex optimization framework.

Contribution

It introduces a novel formulation of conditional graph entropy as an alternating minimization problem, providing a new computational method and theoretical insights.

Findings

Developed an iterative algorithm for computing conditional graph entropy.

Established a connection between conditional graph entropy and convex optimization over a convex body.

Provided a dual problem for optimality checking and error bounds.

Abstract

Conditional graph entropy is known to be the minimal rate for a natural functional compression problem with side information at the receiver. In this paper we show that it can be formulated as an alternating minimization problem, which gives rise to a simple iterative algorithm for numerically computing (conditional) graph entropy. This also leads to a new formula which shows that conditional graph entropy is part of a more general framework: the solution of an optimization problem over a convex corner. In the special case of graph entropy (i.e., unconditioned version) this was known due to Csisz\'ar, K\"orner, Lov\'asz, Marton, and Simonyi. In that case the role of the convex corner was played by the so-called vertex packing polytope. In the conditional version it is a more intricate convex body but the function to minimize is the same. Furthermore, we describe a dual problem that leads to an optimality check and an error bound for the iterative algorithm.