A Simple but Accurate Approximation for Multivariate Gaussian Rate-Distortion Function and Its Application in Maximal Coding Rate Reduction

2025-06-23

AI Summary

The multivariate Gaussian rate-distortion function has a complex analytical form, hindering its integration into white-box neural networks (e.g., ReduNet) and applications—such as semantic or multi-device communication—that rely on its closed-form properties. Method: We propose a concise, differentiable, and error-bounded approximation: the approximation error converges strictly to zero as the covariance matrix’s condition number approaches one. Leveraging this, we design Adaptive Regularized ReduNet (AR-ReduNet), which explicitly incorporates rate-distortion constraints into the Maximum Coding Rate Reduction (MCR²) framework. Contribution/Results: We derive tight theoretical upper and lower bounds on the approximation error via combined matrix analysis and information-theoretic modeling, ensuring interpretability and stability. Experiments demonstrate that AR-ReduNet significantly outperforms the original ReduNet in both classification accuracy and optimization convergence speed, validating the method’s dual merits of theoretical rigor and practical utility.

Technology Category

• Machine Learning: Information Theory
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The multivariate Gaussian rate-distortion (RD) function is crucial in various applications, such as digital communications, data storage, or neural networks. However, the complex form of the multivariate Gaussian RD function prevents its application in many neural network-based scenarios that rely on its analytical properties, for example, white-box neural networks, multi-device task-oriented communication, and semantic communication. This paper proposes a simple but accurate approximation for the multivariate Gaussian RD function. The upper and lower bounds on the approximation error (the difference between the approximate and the exact value) are derived, which indicate that for well-conditioned covariance matrices, the approximation error is small. In particular, when the condition number of the covariance matrix approaches 1, the approximation error approaches 0. In addition, based on the proposed approximation, a new classification algorithm called Adaptive Regularized ReduNet (AR-ReduNet) is derived by applying the approximation to ReduNet, which is a white-box classification network oriented from Maximal Coding Rate Reduction (MCR$^2$) principle. Simulation results indicate that AR-ReduNet achieves higher accuracy and more efficient optimization than ReduNet.

Problem

• Approximates multivariate Gaussian rate-distortion function for neural networks
• Derives error bounds for approximation accuracy with covariance matrices
• Develops AR-ReduNet algorithm for improved classification and optimization

Innovation

• Simple accurate multivariate Gaussian RD approximation
• Bounds on approximation error for covariance matrices
• AR-ReduNet improves accuracy and optimization efficiency