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Nonasymptotic Oblivious Relaying and Variable-Length Noisy Lossy Source Coding

Yanxiao Liu, Sepehr Heidari Advary, Cheuk Ting Li

TL;DR

This work addresses finite-blocklength achievability for the oblivious relay (information bottleneck) channel, wherein a rate-limited description $W$ of the channel output reaches the decoder rather than the output itself. It develops nonasymptotic, second-order characterizations for both fixed-length and variable-length relay descriptions by leveraging nonasymptotic noisy lossy source coding results, the strong functional representation lemma, and Poisson functional representation, and introduces a novel nonasymptotic variable-length noisy lossy source coding bound. The analysis defines the var-information bottleneck $\mathrm{VIB}_{X\to Y}(\mathsf{C})$ and the conditional var-information bottleneck $\mathrm{CVIB}_{X\to Y}(\mathsf{C})$ to capture finite-blocklength fluctuations; for fixed-length descriptions, the rate scales as $\mathrm{IB}_{X\to Y}(\mathsf{C}) + \sqrt{\mathrm{VIB}(\mathsf{C})/n}\,Q^{-1}(\epsilon) + O(\log n)$, while for variable-length descriptions one obtains a $(1-\epsilon)$–scaling with $\sqrt{(\ln n)/n}$ and $\mathrm{CVIB}(\mathsf{C})$. These results reveal a distinct finite-blocklength behavior between the two description regimes and extend nonasymptotic techniques from noisy lossy source coding to the indirect channel setting, with potential implications for cloud-based radio access networks and related architectures.

Abstract

The information bottleneck channel (or the oblivious relay channel) concerns a channel coding setting where the decoder does not directly observe the channel output. Rather, the channel output is relayed to the decoder by an oblivious relay (which does not know the codebook) via a rate-limited link. The capacity is known to be given by the information bottleneck. We study finite-blocklength achievability results of the channel, where the relay communicates to the decoder via fixed-length or variable-length codes. These two cases give rise to two different second-order versions of the information bottleneck. Our proofs utilize the nonasymptotic noisy lossy source coding results by Kostina and Verdú, the strong functional representation lemma, and the Poisson matching lemma. Moreover, we also give a novel nonasymptotic variable-length noisy lossy source coding result.

Nonasymptotic Oblivious Relaying and Variable-Length Noisy Lossy Source Coding

TL;DR

This work addresses finite-blocklength achievability for the oblivious relay (information bottleneck) channel, wherein a rate-limited description of the channel output reaches the decoder rather than the output itself. It develops nonasymptotic, second-order characterizations for both fixed-length and variable-length relay descriptions by leveraging nonasymptotic noisy lossy source coding results, the strong functional representation lemma, and Poisson functional representation, and introduces a novel nonasymptotic variable-length noisy lossy source coding bound. The analysis defines the var-information bottleneck and the conditional var-information bottleneck to capture finite-blocklength fluctuations; for fixed-length descriptions, the rate scales as , while for variable-length descriptions one obtains a –scaling with and . These results reveal a distinct finite-blocklength behavior between the two description regimes and extend nonasymptotic techniques from noisy lossy source coding to the indirect channel setting, with potential implications for cloud-based radio access networks and related architectures.

Abstract

The information bottleneck channel (or the oblivious relay channel) concerns a channel coding setting where the decoder does not directly observe the channel output. Rather, the channel output is relayed to the decoder by an oblivious relay (which does not know the codebook) via a rate-limited link. The capacity is known to be given by the information bottleneck. We study finite-blocklength achievability results of the channel, where the relay communicates to the decoder via fixed-length or variable-length codes. These two cases give rise to two different second-order versions of the information bottleneck. Our proofs utilize the nonasymptotic noisy lossy source coding results by Kostina and Verdú, the strong functional representation lemma, and the Poisson matching lemma. Moreover, we also give a novel nonasymptotic variable-length noisy lossy source coding result.
Paper Structure (13 sections, 9 theorems, 47 equations, 1 figure)

This paper contains 13 sections, 9 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Works
  3. Information Bottleneck Channel and Oblivious Relaying
  4. Noiseless and Noisy Lossy Source Coding
  5. Channel Simulation, Poisson Functional Representation
  6. Noisy Lossy Source Coding
  7. Information Bottleneck Channel
  8. Variable-length Description
  9. Fixed-length Description
  10. Concluding Remarks
  11. Acknowledgement
  12. Remainder of the Proof of Theorem \ref{['thm:noisy_vl']}
  13. Proof of Theorem \ref{['thm:noisy_vl_2']}

Key Result

Theorem 1

For any $P_{\bar{Z}}$ and $\gamma>0$, there exists a fixed-length code with where $T\sim\mathrm{Unif}(0,1)$ is independent of $Y$, and (We assume $Z{\perp\!\!\!\perp}(X,Y)$ above.)

Figures (1)

  • Figure 1: Information bottleneck channel (or oblivious relaying).

Theorems & Definitions (9)

  • Theorem 1: kostina2016nonasymptotic
  • Theorem 2
  • Theorem 3: kostina2016nonasymptotic
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9