We study the rate-distortion problem for both scalar and vector memoryless heavy-tailed $α$-stable sources ($0 < α< 2$). Using a recently defined notion of ``strength" as a power measure, we derive the rate-distortion function for $α$-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 $α$-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 ($α= 2$) under a quadratic distortion measure.

翻译：本文研究了标量与向量无记忆重尾$α$-稳定信源（$0 < α< 2$）的率失真问题。利用近期定义的“强度”这一功率测度概念，我们推导了在误差强度约束下$α$-稳定信源的率失真函数，并证明该函数与强度-失真比呈对数关系。我们展示了该框架如何为$α$-稳定信源及其他广义重尾信源寻找最优量化器铺平道路。此外，我们研究了高速率标量量化器，证明在误差强度失真测度下均匀量化器是渐近最优的。通过比较均匀高斯量化器与柯西量化器，我们发现为保证相同的量化质量，柯西信源需要更多的表示点。我们的研究结果推广了二次失真测度下高斯信源（$α= 2$）率失真与量化的经典结论。