Refined Asymptotics for Rate-Distortion using Gaussian Codebooks for Arbitrary Sources

TL;DR

This paper extends Lapidoth's analysis of Gaussian codebook rate-distortion problems by deriving refined asymptotics, excess-distortion exponents, and comparing spherical and i.i.d. Gaussian codebooks for arbitrary sources.

Contribution

It provides the second-order, moderate, and large deviation asymptotics for Gaussian codebooks, and compares the performance of spherical and i.i.d. Gaussian codebooks in rate-distortion.

Findings

Dispersions are identical for spherical and i.i.d. Gaussian codebooks.

i.i.d. Gaussian codebooks have larger excess-distortion exponents than spherical ones.

Refined asymptotics are established for stationary, memoryless sources.

Abstract

The rate-distortion saddle-point problem considered by Lapidoth (1997) consists in finding the minimum rate to compress an arbitrary ergodic source when one is constrained to use a random Gaussian codebook and minimum (Euclidean) distance encoding is employed. We extend Lapidoth's analysis in several directions in this paper. Firstly, we consider refined asymptotics. In particular, when the source is stationary and memoryless, we establish the second-order, moderate, and large deviation asymptotics of the problem. Secondly, by "random Gaussian codebook", Lapidoth referred to a collection of random codewords, each of which is drawn independently and uniformly from the surface of an $n$-dimensional sphere. To be more precise, we term this as a spherical codebook. We also consider i.i.d.\ Gaussian codebooks in which each random codeword is drawn independently from a product Gaussian distribution. We derive the second-order, moderate, and large deviation asymptotics when i.i.d.\ Gaussian codebooks are employed. In contrast to the recent work on the channel coding counterpart by Scarlett, Tan and Durisi (2017), the dispersions for spherical and i.i.d.\ Gaussian codebooks are identical. The ensemble excess-distortion exponents for both spherical and i.i.d.\ Gaussian codebooks are established for all rates. Furthermore, we show that the i.i.d.\ Gaussian codebook has a strictly larger excess-distortion exponent than its spherical counterpart for any rate greater than the ensemble rate-distortion function derived by Lapidoth.