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On the Second-Order Achievabilities of Indirect Quadratic Lossy Source Coding

Huiyuan Yang, Xiaojun Yuan

TL;DR

The paper addresses second-order achievability for indirect quadratic lossy source coding with continuous alphabets, focusing on a model where $S=\varphi(X)+W$ and $W$ has finite sixth moment. It derives second-order upper bounds using distortion-tilted information, first for recovering only $S$ and then for jointly recovering $(S,X)$ under a generalized distortion constraint, with bounds matching the form of finite-alphabet results. The approach circumvents type-based methods by exploiting the problem structure and employing Berry–Esseen arguments to obtain a Gaussian-approximation term $\sqrt{k}$. The findings enable accurate finite-blocklength approximations for indirect lossy coding and pave the way toward extensions to Gaussian settings and relaxed distortion models.

Abstract

This paper studies the second-order achievabilities of indirect quadratic lossy source coding for a specific class of source models, where the term "quadratic" denotes that the reconstruction fidelity of the hidden source is quantified by a squared error distortion measure. Specifically, it is assumed that the hidden source $S$ can be expressed as $S = \varphi(X) + W$, where $X$ is the observable source with alphabet $\mathcal{X}$, $\varphi(\cdot)$ is a deterministic function, and $W$ is a random variable independent of $X$, satisfying $\mathbb{E}[W] = 0$, $\mathbb{E}[W^2] > 0$, $\mathbb{E}[W^3] = 0$, and $\mathbb{E}[W^6] < \infty$. Additionally, both the set $\{\varphi(x):\ x \in \mathcal{X} \}$ and the reconstruction alphabet for $S$ are assumed to be bounded. Under the above settings, a second-order achievability bound is established using techniques based on distortion-tilted information. This result is then generalized to the case of indirect quadratic lossy source coding with observed source reconstruction, where reconstruction is required for both the hidden source $S$ and the observable source $X$, and the distortion measure for $X$ is not necessarily quadratic. These obtained bounds are consistent in form with their finite-alphabet counterparts, which have been proven to be second-order tight.

On the Second-Order Achievabilities of Indirect Quadratic Lossy Source Coding

TL;DR

The paper addresses second-order achievability for indirect quadratic lossy source coding with continuous alphabets, focusing on a model where and has finite sixth moment. It derives second-order upper bounds using distortion-tilted information, first for recovering only and then for jointly recovering under a generalized distortion constraint, with bounds matching the form of finite-alphabet results. The approach circumvents type-based methods by exploiting the problem structure and employing Berry–Esseen arguments to obtain a Gaussian-approximation term . The findings enable accurate finite-blocklength approximations for indirect lossy coding and pave the way toward extensions to Gaussian settings and relaxed distortion models.

Abstract

This paper studies the second-order achievabilities of indirect quadratic lossy source coding for a specific class of source models, where the term "quadratic" denotes that the reconstruction fidelity of the hidden source is quantified by a squared error distortion measure. Specifically, it is assumed that the hidden source can be expressed as , where is the observable source with alphabet , is a deterministic function, and is a random variable independent of , satisfying , , , and . Additionally, both the set and the reconstruction alphabet for are assumed to be bounded. Under the above settings, a second-order achievability bound is established using techniques based on distortion-tilted information. This result is then generalized to the case of indirect quadratic lossy source coding with observed source reconstruction, where reconstruction is required for both the hidden source and the observable source , and the distortion measure for is not necessarily quadratic. These obtained bounds are consistent in form with their finite-alphabet counterparts, which have been proven to be second-order tight.

Paper Structure

This paper contains 17 sections, 10 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Indirect Quadratic Lossy Source Coding
  5. $(k, M, d_s, \epsilon)$ Code
  6. Rate-Distortion Function
  7. Distortion-Tilted Information
  8. Indirect Quadratic Lossy Source Coding with Observed Source Reconstruction
  9. Main Results: Second-Order Achievabilities
  10. Proof of Theorem \ref{['theorem_Gaussian_approximation_achi']}
  11. Proof of Theorem \ref{['theorem_Gaussian_approximation_achi']}
  12. Discussion of the Proof
  13. Conclusion
  14. Auxiliary Results
  15. Proof of Lemma \ref{['tackle_V_lemma']}
  16. ...and 2 more sections

Key Result

Theorem 1

(Second-order Achievability for $(k, M, d_s, \epsilon)$ Codes): Fix $0 < \epsilon < 1$ and $d_{s, \min} < d_s < d_{s, \max}$. Under the source model and distortion measure introduced in Section Preliminaries, we have where $f(k)=O(g(k))$ means $\lim\sup_{k \to \infty}\left|f(k)/g(k)\right| < \infty$.

Figures (1)

  • Figure 1: Indirect quadratic lossy source coding with/without observed source reconstruction.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof : Proof
  • Theorem 2
  • proof : Proof
  • Lemma 1
  • proof : Proof
  • Lemma 2
  • proof : Proof
  • ...and 8 more