Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rate-Distortion-Classification Representation Theory for Bernoulli Sources

Nam Nguyen, Thinh Nguyen, Bella Bose

TL;DR

The paper addresses task-oriented lossy compression for a Bernoulli source $X$ with a binary classification variable $S$ linked through a binary symmetric model, under Hamming distortion. It develops a one-shot RDC theory with common randomness, deriving closed-form RDC/DRC characterizations and establishing the distortion–classification (DC) achievable region for a fixed representation $Z$ via a linear program. It then extends to universal encoders by providing computable lower and upper bounds on the minimum rate penalty $\Delta(\Theta)$ for supporting a family of DC operating points, including the endpoint analysis and linear-program-based bounds for $R^{(\infty)}(\Theta(R))$. Numerical results illustrate the achievable RDC/DRC regions and the universal curves, offering analytically tractable benchmarks for universality in task-oriented lossy compression of binary sources.

Abstract

We study task-oriented lossy compression through the lens of rate-distortion-classification (RDC) representations. The source is Bernoulli, the distortion measure is Hamming, and the binary classification variable is coupled to the source via a binary symmetric model. Building on the one-shot common-randomness formulation, we first derive closed-form characterizations of the one-shot RDC and the dual distortion-rate-classification (DRC) tradeoffs. We then use a representation-based viewpoint and characterize the achievable distortion-classification (DC) region induced by a fixed representation by deriving its lower boundary via a linear program. Finally, we study universal encoders that must support a family of DC operating points and derive computable lower and upper bounds on the minimum asymptotic rate required for universality, thereby yielding bounds on the corresponding rate penalty. Numerical examples are provided to illustrate the achievable regions and the resulting universal RDC/DRC curves.

Rate-Distortion-Classification Representation Theory for Bernoulli Sources

TL;DR

The paper addresses task-oriented lossy compression for a Bernoulli source with a binary classification variable linked through a binary symmetric model, under Hamming distortion. It develops a one-shot RDC theory with common randomness, deriving closed-form RDC/DRC characterizations and establishing the distortion–classification (DC) achievable region for a fixed representation via a linear program. It then extends to universal encoders by providing computable lower and upper bounds on the minimum rate penalty for supporting a family of DC operating points, including the endpoint analysis and linear-program-based bounds for . Numerical results illustrate the achievable RDC/DRC regions and the universal curves, offering analytically tractable benchmarks for universality in task-oriented lossy compression of binary sources.

Abstract

We study task-oriented lossy compression through the lens of rate-distortion-classification (RDC) representations. The source is Bernoulli, the distortion measure is Hamming, and the binary classification variable is coupled to the source via a binary symmetric model. Building on the one-shot common-randomness formulation, we first derive closed-form characterizations of the one-shot RDC and the dual distortion-rate-classification (DRC) tradeoffs. We then use a representation-based viewpoint and characterize the achievable distortion-classification (DC) region induced by a fixed representation by deriving its lower boundary via a linear program. Finally, we study universal encoders that must support a family of DC operating points and derive computable lower and upper bounds on the minimum asymptotic rate required for universality, thereby yielding bounds on the corresponding rate penalty. Numerical examples are provided to illustrate the achievable regions and the resulting universal RDC/DRC curves.
Paper Structure (12 sections, 7 theorems, 73 equations, 9 figures)

This paper contains 12 sections, 7 theorems, 73 equations, 9 figures.

Table of Contents

  1. Introduction
  2. RDC Tradeoff with Common Randomness
  3. Problem Formulation
  4. RDC/DRC Functions for Bernoulli Sources
  5. DC Achievable Region
  6. Universal Encoder Representation
  7. Conclusion
  8. Proof of Theorem \ref{['thm:oneshot_random_simplified']}
  9. Proof of Theorem \ref{['Oneshot_Bernoulli_radom_RDC']}
  10. Proof of Theorem \ref{['Oneshot_Bernoulli_radom_DRC']}
  11. Proof of Theorem \ref{['theo:lowerBoundary']}
  12. Proof of Theorem \ref{['theo:ratePenaltyBound']}

Key Result

Theorem 1

Suppose that $\Theta$ contains a single pair $(D, C)$. For a source $X \sim p_X$ and an associated classification variable $S \sim p_S$, the one-shot RDC function can be written as where $p_{U,X,\hat{X}}=p_U\,p_X\,p_{\hat{X}|U,X}$.

Figures (9)

  • Figure 1: Task-oriented lossy compression framework.
  • Figure 2: One-shot setting with common randomness.
  • Figure 3: System architecture of Theorem \ref{['thm:oneshot_random_simplified']}.
  • Figure 4: Illustration of Theorem \ref{['Oneshot_Bernoulli_radom_RDC']}. $R^{(B)}(D,C)$ versus $D$ with given $C=0.8$, $q_X = 0.3$, $q_{S_1} = 0.2$.
  • Figure 5: RDC curves for a fixed $C$: $R^{(B)}(D,C)$ and $R^{(\infty)}(D,C)$ versus $D$, for $C=0.9$, $q_X=0.3$, and $q_{S_1}=0.2$.
  • ...and 4 more figures

Theorems & Definitions (19)

  • Definition 1: One-Shot RDC Function
  • Definition 2: Asymptotic RDC Function
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Definition 3
  • Theorem 4
  • proof
  • ...and 9 more