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arXiv · 2608.24355

Universal Random Coding for Successive Refinement of Individual Sequences Based on Lempel-Ziv Complexity

Neri Merhav
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Abstract

This work extends an earlier proposed universal random-coding ensemble for sample-wise lossy compression of individual sequences to the two-stage (successive-refinement) setting --- an extension that is not quite straightforward, as explained in the sequel. The construction has three layers. First, a complete, self-contained direct/converse match in the language of type classes and typical sequences, a two-stage analogue of Rimoldi's classical characterization (applied in the superalphabet of blocks). Second, a realization of the same rates via a random-coding scheme built from the Lempel--Ziv (LZ78) complexity rather than the type-class uniform measure the first layer relies on, since an LZ-based ensemble is universal across source statistics and distortion measures, carries no block-length parameter in the codebook itself, and is directly implementable. Third, we give a fully type-free achievability construction (no joint types of $\ell$-vectors appear anywhere in its search criterion) and show it matches the same converse exactly, reaching every point of the achievable region: the same converse from the first layer already binds any code, type-free schemes included, so no separate argument is needed to certify the match. We also describe (without proof) how to extend the type-free construction to any fixed number $r>2$ of stages. Finally, we show this paper's achievable region is a superset of the one obtained by a companion paper, which is based on a finite-state modeling approach to a similar two-stage problem, extending a known single-stage domination result to two stages.

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Neri Merhav. 2026-08-25. Universal Random Coding for Successive Refinement of Individual Sequences Based on Lempel-Ziv Complexity. https://arxiv.org/abs/2608.24355

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