# Strong Functional Representation Lemma and Applications to Coding Theorems

## TL;DR

This paper introduces a strong functional representation lemma (SFRL) that provides bounds on the information needed to represent one random variable as a function of another and an independent variable, with applications to various coding theorems.

## Contribution

The paper presents the SFRL, a novel lemma that improves bounds in information representation and simplifies proofs in channel simulation and coding theorems.

## Key findings

- Bound on the rate for one-shot exact channel simulation.
- New achievability results for lossy source coding and multiple description coding.
- Simplified proof of the Gelfand-Pinsker theorem using SFRL.

## Abstract

This paper shows that for any random variables $X$ and $Y$, it is possible to represent $Y$ as a function of $(X,Z)$ such that $Z$ is independent of $X$ and $I(X;Z|Y)\le\log(I(X;Y)+1)+4$ bits. We use this strong functional representation lemma (SFRL) to establish a bound on the rate needed for one-shot exact channel simulation for general (discrete or continuous) random variables, strengthening the results by Harsha et al. and Braverman and Garg, and to establish new and simple achievability results for one-shot variable-length lossy source coding, multiple description coding and Gray-Wyner system. We also show that the SFRL can be used to reduce the channel with state noncausally known at the encoder to a point-to-point channel, which provides a simple achievability proof of the Gelfand-Pinsker theorem.