Functional Epsilon Entropy

TL;DR

This paper studies the minimal communication rate needed for a receiver to compute a function of combined data with fidelity constraints, introducing hypergraph entropy and practical coding schemes, and analyzing rate discontinuities.

Contribution

It establishes the minimum rate as hypergraph entropy, designs practical codes, and explores rate discontinuities and approximate functions in the context of computing with distortion.

Findings

Minimum rate equals hypergraph entropy.

Practical coding schemes are developed.

Rate may be discontinuous with fidelity constraints.

Abstract

We consider the problem of coding for computing with maximal distortion, where the sender communicates with a receiver, which has its own private data and wants to compute a function of their combined data with some fidelity constraint known to both agents. We show that the minimum rate for this problem is equal to the conditional entropy of a hypergraph and design practical codes for the problem. Further, the minimum rate of this problem may be a discontinuous function of the fidelity constraint. We also consider the case when the exact function is not known to the sender, but some approximate function or a class to which the function belongs is known and provide efficient achievable schemes.