A Framework for Robust Lossy Compression of Heavy-Tailed Sources

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 showcase how our framework paves the way to finding optimal quantizers for $\alpha$-stable sources and more generally to heavy-tailed ones. In addition, we study high-rate scalar quantizers and show that uniform ones are asymptotically optimal under the strength 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 rate-distortion and quantization results of Gaussian sources ($\alpha = 2$) under a quadratic distortion measure.