Successive Refinement for Lossy Compression of Individual Sequences

TL;DR

The paper addresses successive refinement for lossy compression of deterministic sequences using a two-stage scheme with finite-state encoders, enforcing distortions $d_1(x^n,\hat{x}^n)\le nD_1$ and $d_2(x^n,\tilde{x}^n)\le nD_2$. It derives outer bounds on the achievable rate region based on empirical LZ complexities, and shows that a conceptually simple LZ-based achievability matches the outer bounds asymptotically as $n\to\infty$ for fixed state count $q$, with a similar extension to any fixed number of stages. The framework is then generalized to the two-description setting, giving inner bounds analogous to El Gamal-Cover and Zhang-Berger schemes for memoryless sources, adapted to the deterministic, finite-state regime. Overall, the work links finite-state coding for individual sequences with LZ complexity and provides a coherent outer/inner bound structure that parallels classical probabilistic-rate-distortion results, while enabling multi-stage and multi-description scalability.

Abstract

We consider the problem of successive-refinement coding for lossy compression of individual sequences, namely, compression in two stages, where in the first stage, a coarse description at a relatively low rate is sent from the encoder to the decoder, and in the second stage, additional coding rate is allocated in order to refine the description and thereby improve the reproduction. Our main result is in establishing outer bounds (converse theorems) for the rate region where we limit the encoders to be finite-state machines in the spirit of Ziv and Lempel's 1978 model.The matching achievability scheme is conceptually straightforward. We also consider the more general multiple description coding problem on a similar footing and propose achievability schemes that are analogous to the well-known El Gamal-Cover and the Zhang-Berger achievability schemes of memoryless sources and additive distortion measures.