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A Theory of Universal Rate-Distortion-Classification Representations for Lossy Compression

Nam Nguyen, Thinh Nguyen, Bella Bose

TL;DR

The paper addresses multi-objective lossy compression by extending RD theory with perception and classification, proposing universal representations that fix the encoder and reuse decoders to achieve diverse distortion-classification tradeoffs. It proves that for Gaussian sources under MSE, a single fixed encoder can realize the entire distortion-classification region without rate penalty, and it provides a generalized characterization for arbitrary sources using MMSE and Wasserstein concepts, including an asymptotic equivalence R^(∞)(Θ) = R(Θ). Empirically, universal encoders trained with Wasserstein regularization perform comparably to task-specific models on MNIST and SVHN, validating practicality for multi-task compression. The results offer a scalable approach to multi-objective compression where updating decoders suffices to meet varying downstream requirements, reducing design and training burden while preserving performance. These findings advance the deployment of versatile, task-aware compression systems in real-world applications with strict perceptual and analytical requirements.

Abstract

In lossy compression, Blau and Michaeli [5] introduced the information rate-distortion-perception (RDP) function, extending traditional rate-distortion theory by incorporating perceptual quality. More recently, this framework was expanded by defining the rate-distortion-perception-classification (RDPC) function, integrating multi-task learning that jointly optimizes generative tasks such as perceptual quality and classification accuracy alongside reconstruction tasks [28]. To that end, motivated by the concept of a universal RDP encoder introduced in [34], we investigate universal representations that enable diverse distortion-classification tradeoffs through a single fixed encoder combined with multiple decoders. Specifically, theoretical analysis and numerical experiment demonstrate that for the Gaussian source under mean squared error (MSE) distortion, the entire distortion-classification tradeoff region can be achieved using one universal encoder. In addition, this paper characterizes achievable distortion-classification regions for fixed universal representations in general source distributions, identifying conditions that ensure minimal distortion penalty when reusing encoders across varying tradeoff points. Experimental results using MNIST and SVHN datasets validate our theoretical insights, showing that universal encoders can obtain distortion performance comparable to task-specific encoders, thus supporting the practicality and effectiveness of our proposed universal representations.

A Theory of Universal Rate-Distortion-Classification Representations for Lossy Compression

TL;DR

The paper addresses multi-objective lossy compression by extending RD theory with perception and classification, proposing universal representations that fix the encoder and reuse decoders to achieve diverse distortion-classification tradeoffs. It proves that for Gaussian sources under MSE, a single fixed encoder can realize the entire distortion-classification region without rate penalty, and it provides a generalized characterization for arbitrary sources using MMSE and Wasserstein concepts, including an asymptotic equivalence R^(∞)(Θ) = R(Θ). Empirically, universal encoders trained with Wasserstein regularization perform comparably to task-specific models on MNIST and SVHN, validating practicality for multi-task compression. The results offer a scalable approach to multi-objective compression where updating decoders suffices to meet varying downstream requirements, reducing design and training burden while preserving performance. These findings advance the deployment of versatile, task-aware compression systems in real-world applications with strict perceptual and analytical requirements.

Abstract

In lossy compression, Blau and Michaeli [5] introduced the information rate-distortion-perception (RDP) function, extending traditional rate-distortion theory by incorporating perceptual quality. More recently, this framework was expanded by defining the rate-distortion-perception-classification (RDPC) function, integrating multi-task learning that jointly optimizes generative tasks such as perceptual quality and classification accuracy alongside reconstruction tasks [28]. To that end, motivated by the concept of a universal RDP encoder introduced in [34], we investigate universal representations that enable diverse distortion-classification tradeoffs through a single fixed encoder combined with multiple decoders. Specifically, theoretical analysis and numerical experiment demonstrate that for the Gaussian source under mean squared error (MSE) distortion, the entire distortion-classification tradeoff region can be achieved using one universal encoder. In addition, this paper characterizes achievable distortion-classification regions for fixed universal representations in general source distributions, identifying conditions that ensure minimal distortion penalty when reusing encoders across varying tradeoff points. Experimental results using MNIST and SVHN datasets validate our theoretical insights, showing that universal encoders can obtain distortion performance comparable to task-specific encoders, thus supporting the practicality and effectiveness of our proposed universal representations.

Paper Structure

This paper contains 26 sections, 12 theorems, 140 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Rate-Distortion-Classification Representations
  4. Gaussian Source Case
  5. Convexity and Bounds of the Distortion-Classification-Rate Function for a General Source
  6. Universal Representations
  7. Definitions
  8. Main Results
  9. Experimental Results
  10. Illustration of No Rate Penalty for a Gaussian Source
  11. Universal Representation for Lossy Compression
  12. Training
  13. Results
  14. Conclusion
  15. Theoretical Results
  16. ...and 11 more sections

Key Result

Theorem 1

Wang2024 Let $X\sim \mathcal{N}(\mu_X,\sigma_X^2)$ be a Gaussian source and $S\sim \mathcal{N}(\mu_S,\sigma_S^2)$ be an associated classification variable, with a covariance of $\text{Cov}(X,S) = \theta_1$. The problem is feasible if $C \geq \frac{1}{2} \log\left(1 - \frac{\theta_1^2}{\sigma_S^2 \si where $\rho = \frac{\theta_1}{\sigma_S \sigma_X}$ represents the correlation coefficient between $X

Figures (11)

  • Figure 1: Illustration of task-oriented lossy compression framework.
  • Figure 2: Illustration of the information rate-distortion-classification function of a Gaussian source.
  • Figure 3: Illustration of the distortion-classification-rate functions: \ref{['fig:DCR']} shows the DCR function at a fixed rate, and \ref{['fig:DCR_Mutltiple_Rate']} shows how the function changes across multiple rates.
  • Figure 4: Bounding the classification-distortion-rate function: the gap between $D_{\min}$ and $D_{\max}$ quantifies the additional distortion required to achieve the minimum classification loss. For the MSE distortion, Theorem \ref{['Theorem_CDR_Bound']} establishes that this increase is bounded by a factor of 2, corresponding to a 3 dB drop in PSNR.
  • Figure 5: Illustration of the universal representation framework.
  • ...and 6 more figures

Theorems & Definitions (36)

  • Definition 1: Information Rate-Distortion-Classification Function
  • Theorem 1: Information Rate-Distortion-Classification Function for a Gaussian Source
  • proof
  • Theorem 2: Information Distortion-Classification-Rate Function for a Gaussian Source
  • proof
  • Theorem 3
  • proof
  • Theorem 4: Bound on the Classification-Distortion-Rate Function
  • proof
  • Definition 2: Information Universal Rate-Distortion-Classification Function
  • ...and 26 more