A Framework for Robust Lossy Compression of Heavy-Tailed Sources

TL;DR

This paper develops a rate-distortion framework for heavy-tailed $ ext{alpha}$-stable sources, deriving the rate-distortion function, analyzing quantizer optimality, and comparing different source distributions.

Contribution

It introduces a new rate-distortion analysis for heavy-tailed sources using the notion of strength, extending classical Gaussian results to $ ext{alpha}$-stable sources.

Findings

Rate-distortion function is logarithmic in the strength-to-distortion ratio.

Uniform scalar quantizers are asymptotically optimal for heavy-tailed sources.

Cauchy sources require more points than Gaussian sources for the same quality.

Abstract

We study the rate-distortion problem for both scalar and vector memoryless heavy-tailed $\alpha$-stable sources ($0 < \alpha < 2$). Using a recently defined notion of ``strength" as a power measure, we derive the rate-distortion function for $\alpha$-stable sources subject to a constraint on the strength of the error and show it to be logarithmic in the strength-to-distortion ratio. We show how our framework paves the way for finding optimal quantizers for $\alpha$-stable sources and other general heavy-tailed ones. In addition, we study high-rate scalar quantizers and show that uniform ones are asymptotically optimal under the error-strength distortion measure. We compare uniform Gaussian and Cauchy quantizers and show that more representation points for the Cauchy source are required to guarantee the same quantization quality. Our findings generalize the well-known results of rate-distortion and quantization of Gaussian sources ($\alpha = 2$) under a quadratic distortion measure.