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One-Shot Wyner-Ziv Compression of a Uniform Source

Oğuzhan Kubilay Ülger, Elza Erkip

TL;DR

This work addresses the one‑shot Wyner–Ziv problem for a uniform source under $L_1$ distortion with decoder‑only side information. It derives the point‑to‑point entropy–distortion function $H^U(\Delta)$ and provides computable bounds for decoder‑only SI under both quantized ($H_{SI}^q(\Delta)$) and noisy ($H_{SI}^n(\Delta)$) side information, using interval‑partition and grouping strategies for achievability and convex‑envelope arguments for converses. The results show that the bounds tighten at higher rates and with stronger source–SI correlation, and they elucidate the impact of SI quantization vs. noise on the entropy‑distortion trade‑offs. These insights have practical relevance for finite‑blocklength distributed compression and potentially guide the design of decoder‑aware neural compressors for low‑dimensional features embedded in high‑dimensional data.

Abstract

In this paper, we consider the one-shot version of the classical Wyner-Ziv problem where a source is compressed in a lossy fashion when only the decoder has access to a correlated side information. Following the entropy-constrained quantization framework, we assume a scalar quantizer followed by variable length entropy coding. We consider compression of a uniform source, motivated by its role in the compression of processes with low-dimensional features embedded within a high-dimensional ambient space. We find upper and lower bounds to the entropy-distortion functions of the uniform source for quantized and noisy side information, and illustrate tightness of the bounds at high compression rates.

One-Shot Wyner-Ziv Compression of a Uniform Source

TL;DR

This work addresses the one‑shot Wyner–Ziv problem for a uniform source under distortion with decoder‑only side information. It derives the point‑to‑point entropy–distortion function and provides computable bounds for decoder‑only SI under both quantized () and noisy () side information, using interval‑partition and grouping strategies for achievability and convex‑envelope arguments for converses. The results show that the bounds tighten at higher rates and with stronger source–SI correlation, and they elucidate the impact of SI quantization vs. noise on the entropy‑distortion trade‑offs. These insights have practical relevance for finite‑blocklength distributed compression and potentially guide the design of decoder‑aware neural compressors for low‑dimensional features embedded in high‑dimensional data.

Abstract

In this paper, we consider the one-shot version of the classical Wyner-Ziv problem where a source is compressed in a lossy fashion when only the decoder has access to a correlated side information. Following the entropy-constrained quantization framework, we assume a scalar quantizer followed by variable length entropy coding. We consider compression of a uniform source, motivated by its role in the compression of processes with low-dimensional features embedded within a high-dimensional ambient space. We find upper and lower bounds to the entropy-distortion functions of the uniform source for quantized and noisy side information, and illustrate tightness of the bounds at high compression rates.
Paper Structure (8 sections, 2 theorems, 42 equations, 2 figures)

This paper contains 8 sections, 2 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Main Results
  4. Quantized Side Information
  5. Noisy Side Information
  6. Illustrations and Discussion
  7. Proof of Theorem \ref{['thm:nconv']}
  8. Proof of Corollary \ref{['cor:n']}

Key Result

Corollary 1

The conditional/decoder-only SI entropy--distortion function for the uniform source with quantized SI in Theorem thm:qach can be upper and lower bounded as where $H^U(\cdot)$ is the entropy--distortion function of the uniform source given in Theorem ThmPtP and $\Breve{H}^U(\cdot)$ is its the convex envelope.

Figures (2)

  • Figure 1: One-shot lossy compression systems considered in this paper: (a) point-to-point, (b) decoder-only side information, (c) both encoder and decoder side information.
  • Figure 2: Upper and lower bounds of entropy--distortion trade-off for quantized and noisy SI case for (a) $K=1/\alpha=4$ and (b) $K=1/\alpha=32$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • proof
  • Corollary 1
  • proof
  • proof
  • ...and 3 more