On the rate distortion function of Bernoulli Gaussian sequences

TL;DR

This paper investigates the rate distortion function of Bernoulli-Gaussian sequences, introducing a novel duality technique that tightens bounds and enhances understanding of sparse signal compression.

Contribution

The paper presents a new method for deriving lower bounds on the rate distortion function, improving previous bounds especially for small distortion and sparse signals.

Findings

Improved lower bound on rate distortion function by p log2(1/p) for small D

Demonstrated near-tight bounds for sparse signals as p approaches zero

Established duality between channel coding and lossy source coding in this context

Abstract

In this paper, we study the rate distortion function of the i.i.d sequence of multiplications of a Bernoulli $p$ random variable and a gaussian random variable $\sim N(0,1)$. We use a new technique in the derivation of the lower bound in which we establish the duality between channel coding and lossy source coding in the strong sense. We improve the lower bound on the rate distortion function over the best known lower bound by $p\log_2\frac{1}{p}$ if distortion $D$ is small. This has some interesting implications on sparse signals where $p$ is small since the known gap between the lower and upper bound is $H(p)$. This improvement in the lower bound shows that the lower and upper bounds are almost identical for sparse signals with small distortion because $\lim\limits_{p\to 0}\frac{p\log_2\frac{1}{p}}{H(p)}=1$.