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

TL;DR

This paper introduces a simple, accurate approximation for the multivariate Gaussian rate-distortion function, enabling its use in neural network applications, and develops a new classification algorithm based on this approximation.

Contribution

It provides a novel approximation with error bounds for the Gaussian RD function and applies it to create the AR-ReduNet classification algorithm.

Findings

The approximation is highly accurate for well-conditioned covariance matrices.

AR-ReduNet outperforms ReduNet in accuracy and optimization efficiency.

Error bounds decrease as the covariance matrix condition number approaches 1.

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.