We consider the rate-distortion function for lossy source compression, as well as the channel capacity for error correction, through the lens of distributional robustness. We assume that the distribution of the source or of the additive channel noise is unknown and lies within a Wasserstein-2 ambiguity set of a given radius centered around a specified nominal distribution, and we look for the worst-case asymptotically optimal coding rate over such an ambiguity set. Varying the radius of the ambiguity set allows us to interpolate between the worst-case and stochastic scenarios using probabilistic tools. Our problem setting fits into the paradigm of compound source / channel models introduced by Sakrison and Blackwell, respectively. This paper shows that if the nominal distribution is Gaussian, then so is the worst-case source / noise distribution, and the compound rate-distortion / channel capacity functions admit convex formulations with Linear Matrix Inequality (LMI) constraints. These formulations yield simple closed-form expressions in the scalar case, offering insights into the behavior of Shannon limits with the changing radius of the Wasserstein-2 ambiguity set.

翻译：我们通过分布鲁棒性的视角，研究了有损信源压缩的率失真函数以及纠错信道容量。假设信源或加性信道噪声的分布未知，且位于以给定标称分布为中心、半径确定的Wasserstein-2模糊集内，我们寻求在该模糊集上最坏情况下的渐近最优编码速率。通过改变模糊集的半径，我们可以利用概率工具在最坏情况与随机场景之间进行插值。该问题设定分别符合Sakrison和Blackwell提出的复合信源/信道模型范式。本文证明：若标称分布为高斯分布，则最坏情况下的信源/噪声分布也为高斯分布，且复合率失真/信道容量函数可转化为带线性矩阵不等式约束的凸优化形式。这些凸优化形式在标量情况下可推导出简洁的闭式表达式，从而揭示香农极限随Wasserstein-2模糊集半径变化的规律。