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Universal Rate-Distortion-Classification Representations for Lossy Compression

Nam Nguyen, Thuan Nguyen, Thinh Nguyen, Bella Bose

TL;DR

This work tackles multi-task lossy compression by introducing a universal RDC framework where a fixed encoder is paired with task-specific decoders to realize a range of distortion and classification constraints. For Gaussian sources under MSE, the authors prove zero rate penalty, showing that the full distortion-classification region can be achieved with a single encoder; for general sources they characterize the achievable region and bound the penalty via minimum mean-squared error and Wasserstein distance analyses. Theoretical results are complemented by experiments on MNIST that validate the universality concept and demonstrate distortion performance on par with task-specific encoders, aided by Wasserstein-regularized training. The findings support a practical universal representation approach for multi-task compression, enabling flexible deployment without retraining encoders across tasks.

Abstract

In lossy compression, Wang et al. [1] recently introduced the rate-distortion-perception-classification function, which supports multi-task learning by jointly optimizing perceptual quality, classification accuracy, and reconstruction fidelity. Building on the concept of a universal encoder introduced in [2], we investigate the universal representations that enable a broad range of distortion-classification tradeoffs through a single shared encoder coupled with multiple task-specific decoders. We establish, through both theoretical analysis and numerical experiments, that for Gaussian source under mean squared error (MSE) distortion, the entire distortion-classification tradeoff region can be achieved using a single universal encoder. For general sources, we characterize the achievable region and identify conditions under which encoder reuse results in negligible distortion penalty. The experimental result on the MNIST dataset further supports our theoretical findings. We show that universal encoders can obtain distortion performance comparable to task-specific encoders. These results demonstrate the practicality and effectiveness of the proposed universal framework in multi-task compression scenarios.

Universal Rate-Distortion-Classification Representations for Lossy Compression

TL;DR

This work tackles multi-task lossy compression by introducing a universal RDC framework where a fixed encoder is paired with task-specific decoders to realize a range of distortion and classification constraints. For Gaussian sources under MSE, the authors prove zero rate penalty, showing that the full distortion-classification region can be achieved with a single encoder; for general sources they characterize the achievable region and bound the penalty via minimum mean-squared error and Wasserstein distance analyses. Theoretical results are complemented by experiments on MNIST that validate the universality concept and demonstrate distortion performance on par with task-specific encoders, aided by Wasserstein-regularized training. The findings support a practical universal representation approach for multi-task compression, enabling flexible deployment without retraining encoders across tasks.

Abstract

In lossy compression, Wang et al. [1] recently introduced the rate-distortion-perception-classification function, which supports multi-task learning by jointly optimizing perceptual quality, classification accuracy, and reconstruction fidelity. Building on the concept of a universal encoder introduced in [2], we investigate the universal representations that enable a broad range of distortion-classification tradeoffs through a single shared encoder coupled with multiple task-specific decoders. We establish, through both theoretical analysis and numerical experiments, that for Gaussian source under mean squared error (MSE) distortion, the entire distortion-classification tradeoff region can be achieved using a single universal encoder. For general sources, we characterize the achievable region and identify conditions under which encoder reuse results in negligible distortion penalty. The experimental result on the MNIST dataset further supports our theoretical findings. We show that universal encoders can obtain distortion performance comparable to task-specific encoders. These results demonstrate the practicality and effectiveness of the proposed universal framework in multi-task compression scenarios.

Paper Structure

This paper contains 15 sections, 5 theorems, 87 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Rate-Distortion-Classification Representations
  3. Gaussian Source
  4. Universal Representations
  5. Definitions
  6. Main Results
  7. Experimental Results
  8. Gaussian Zero-Rate Penalty
  9. Universal Representation for Lossy Compression
  10. Training
  11. Conclusion
  12. Proof of Theorem \ref{['TheoremDCR_GS']}
  13. Proof of Theorem \ref{['Theorem_gaussian_universality']}
  14. Proof of Theorem \ref{['Theorem_general_universality']}
  15. Proof of Theorem \ref{['Theorem_Quantitative_Results']}

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 (RDC) is feasible if $C \geq \frac{1}{2} \log\left(1 - \frac{\theta_1^2}{\sigma_S where $\rho = \frac{\theta_1}{\sigma_S \sigma_X}$ represents the correlation coefficient between $X

Figures (6)

  • Figure 1: Task-oriented lossy compression framework.
  • Figure 2: The universal representation framework.
  • Figure 3: Universality for a general source. Shown are the boundaries of achievable distortion-classification regions for three representations: the minimal distortion point $(D_1, C_1)$, where $R(D_1, C_1) = R(D_1, \infty)$; the midpoint $(D_2, C_2)$; and the minimal classification loss point $(D_3, C_3)$.
  • Figure 4: CDR functions for a Gaussian source.
  • Figure 5: An illustration of the universal RDC scheme.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • proof
  • Definition 3
  • Definition 4
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 3 more