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Universal Slepian-Wolf Coding Theorem for Individual Sequences by Neri Merhav


Core Concepts
The author establishes a coding theorem and converse theorem for separate encodings and joint decoding of individual sequences using finite-state machines, emphasizing the universal decoder's properties.
Abstract
The content discusses the Universal Slepian-Wolf Coding Theorem for individual sequences. It introduces a coding scheme with separate encodings and joint decoding using finite-state machines. The achievable rate region is characterized in terms of LZ complexities, with an emphasis on the universal decoder's role. Draper's incremental SW coding scheme is modified to achieve the universally attainable rate region. In-depth discussions include the theoretical framework, notation conventions, problem formulation, objectives, and background information related to LZ complexities. The paper presents a comprehensive analysis of achievable rate regions and proposes a modified incremental coding scheme for individual sequences.
Stats
ρ(x) = lim sup k→∞ lim sup n→∞ k/n Σi=0 ρLZ(xi_k+1) ρ(x|y) = lim sup k→∞ lim sup n→∞ k/n Σi=0 ρLZ(xi_k+1|yi_k+1) ρ(y|x) = lim sup k→∞ lim sup n→∞ k/n Σi=0 ρLZ(yi_k+1|xi_k+1) Bℓ(ǫ) ≤ 2^ℓQ(ǫ) Q(ǫ) = h2(ǫ) + ǫ log(αβ - 1) δ(ǫ) = h2(ǫ) + ǫ log(αβ - 1)
Quotes
"The achievable rate region is characterized in terms of LZ complexities." "Draper proposed a universal, incremental variable-rate coding scheme that can be implemented provided that a low-rate reliable feedback link is available."

Key Insights Distilled From

by Neri Merhav at arxiv.org 03-13-2024

https://arxiv.org/pdf/2403.07409.pdf
Universal Slepian-Wolf coding for individual sequences

Deeper Inquiries

How does the proposed modified incremental coding scheme compare to existing methods

The proposed modified incremental coding scheme in the context provided builds upon Draper's universal incremental SW coding scheme for memoryless sources. One key difference is that it adapts this scheme to individual sequences using finite-state encoders, allowing for separate encoding and joint decoding of two correlated sequences. This modification enables the achievement of a universally attainable rate region by adjusting coding rates dynamically based on the compressibilities of the source sequences. Compared to existing methods, this modified scheme offers a more flexible and adaptive approach to encoding and decoding individual sequences. By incorporating elements from Draper's universal incremental coding scheme and tailoring them to suit the specific requirements of individual sequence compression, it provides a framework for achieving efficient compression while considering unknown source statistics.

What are the implications of not having an exact corresponding chain rule for LZ complexities

The absence of an exact corresponding chain rule for LZ complexities poses certain implications in information theory and communication systems. In traditional settings like probabilistic scenarios with discrete memoryless sources, having a chain rule allows for easy decomposition of joint complexities into conditional complexities, facilitating efficient encoding and decoding strategies. In cases where such a chain rule does not exist, as seen in the context described above with LZ complexities for individual sequences, it can complicate the design and analysis of coding schemes. Without a straightforward way to decompose joint complexity into conditional complexities or vice versa, achieving optimal compression rates may require more intricate approaches or approximations. However, despite this limitation, establishing relationships between different types of LZ complexities in an asymptotic sense can still provide valuable insights into achievable rate regions and guide the development of effective coding schemes tailored to specific scenarios.

How does the concept of universal decoding metrics apply in different communication settings

Universal decoding metrics play a crucial role in various communication settings by providing adaptive mechanisms that optimize decoding performance under uncertainty or varying conditions. These metrics are designed to minimize errors or distortion during data transmission while accommodating unknown source statistics or channel characteristics. In different communication contexts, such as Slepian-Wolf (SW) coding for individual sequences using finite-state machines as discussed above: Universal decoding metrics help achieve almost lossless compression by dynamically adjusting code rates based on empirical entropy measures. They enable decoders to efficiently decode compressed data even when source statistics are unknown. The use of these metrics ensures reliable communication over noisy channels without requiring prior knowledge about signal properties. Overall, universal decoding metrics enhance robustness and adaptability in communication systems by optimizing performance under diverse conditions.
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