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Pointwise Redundancy in One-Shot Lossy Compression via Poisson Functional Representation

Cheuk Ting Li

TL;DR

This work analyzes pointwise redundancy in one-shot, variable-length lossy compression using the Poisson functional representation. It constructs one-shot schemes with positive-integer descriptions and derives bounds on several notions of pointwise redundancy (PRR, PSR, PSDR), including tail bounds and expectations tied to the rate-distortion function $R(D)$ and the $d$-tilted information, while also extending the framework to the one-shot lossy Gray-Wyner system. The results unify the encoding-length distribution, provide explicit overheads for prefix-free and non-prefix-free schemes, and establish bounds that connect information densities with coding length under distortion constraints. The paper thus offers non-asymptotic guarantees for encoding length close to $R(D)$ and paves the way for a pointwise redundancy theory in more complex networks like Gray-Wyner. These insights have potential practical impact for non-asymptotic coding in lossy compression and networked source coding scenarios.

Abstract

We study different notions of pointwise redundancy in variable-length lossy source coding. We present a construction of one-shot variable-length lossy source coding schemes using the Poisson functional representation, and give bounds on its pointwise redundancy for various definitions of pointwise redundancy. This allows us to describe the distribution of the encoding length in a precise manner. We also generalize the result to the one-shot lossy Gray-Wyner system.

Pointwise Redundancy in One-Shot Lossy Compression via Poisson Functional Representation

TL;DR

This work analyzes pointwise redundancy in one-shot, variable-length lossy compression using the Poisson functional representation. It constructs one-shot schemes with positive-integer descriptions and derives bounds on several notions of pointwise redundancy (PRR, PSR, PSDR), including tail bounds and expectations tied to the rate-distortion function and the -tilted information, while also extending the framework to the one-shot lossy Gray-Wyner system. The results unify the encoding-length distribution, provide explicit overheads for prefix-free and non-prefix-free schemes, and establish bounds that connect information densities with coding length under distortion constraints. The paper thus offers non-asymptotic guarantees for encoding length close to and paves the way for a pointwise redundancy theory in more complex networks like Gray-Wyner. These insights have potential practical impact for non-asymptotic coding in lossy compression and networked source coding scenarios.

Abstract

We study different notions of pointwise redundancy in variable-length lossy source coding. We present a construction of one-shot variable-length lossy source coding schemes using the Poisson functional representation, and give bounds on its pointwise redundancy for various definitions of pointwise redundancy. This allows us to describe the distribution of the encoding length in a precise manner. We also generalize the result to the one-shot lossy Gray-Wyner system.
Paper Structure (8 sections, 9 theorems, 35 equations)

This paper contains 8 sections, 9 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Expected Length
  4. Notions of Pointwise Redundancy
  5. Pointwise Redundancy without Prefix-free Condition
  6. Pointwise Redundancy with Prefix-free Condition
  7. Lossy Gray-Wyner System
  8. Acknowledgement

Key Result

Theorem 1

Fix any $P_{X}$, $P_{Y|X}$ and $Q_{Y}$ satisfying $P_{Y|X}(\cdot|x)\ll Q_{Y}$ for $P_{X}$-almost all $x$'s. Consider any finite collection of functions $\psi_{i}:\mathcal{X}\times\mathcal{Y}\times\mathbb{Z}_{>0}\to\mathbb{R}$ that are nondecreasing in the third argument (i.e., $\psi_{i}(x,y,k)$ is n for $i=1,\ldots,\ell$, where $(X,Y)\sim P_{X}P_{Y|X}$, and $J\in\mathbb{Z}_{>0}$ is distributed as

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2: Poisson functional representation li2021unified
  • Corollary 3: Variable-length lossy compression
  • Corollary 4: Pointwise redundancy w/o prefix-free condition
  • Corollary 5: PSDR w/o prefix-free condition
  • Corollary 6: Pointwise redundancy w/o prefix-free condition
  • Corollary 7: Pointwise redundancy with prefix-free condition
  • Corollary 8: PSDR with prefix-free condition
  • Theorem 9