We provide a unified method for constructing explicit distributions which are difficult for restricted models of computation to generate. Our constructions are based on a new notion of robust extractors, which are extractors that remain sound even when a small number of points violate the min-entropy constraint. Using such objects, we show that for a broad range of sampling models (e.g., low-depth circuits, small-space sources, etc.), every output of the model has distance $1 - o(1)$ from our target distribution, qualitatively recovering essentially all previously known hardness results. Our work extends that of Viola (SICOMP '14), who developed an earlier unified framework based on traditional extractors to rule out sampling with very small error. As a further application of our technique, we leverage a recent extractor construction of Chattopadhyay, Goodman, and Gurumukhani (ITCS '24) to present the first explicit distribution with distance $1 - o(1)$ from the output of any low-degree $\mathbb{F}_2$-polynomial source. We also describe a potential avenue toward proving a similar hardness result for $\mathsf{AC^0}[\oplus]$ circuits.

翻译：我们提出了一种构造显式分布的统方法，这些分布对于受限计算模型而言难以生成。我们的构造基于一种新的鲁棒提取器概念，这类提取器即使在少量数据点违反最小熵约束的情况下仍保持可靠性。利用此类对象，我们证明：对于广泛采样模型（例如低深度电路、小空间源等），模型的每个输出与目标分布的距离为 $1 - o(1)$，本质上恢复了先前已知的所有硬度结果。本研究扩展了Viola (SICOMP '14) 的成果，他此前基于传统提取器建立了排除极小误差采样的统一框架。作为技术的进一步应用，我们借助Chattopadhyay、Goodman和Gurumukhani (ITCS '24) 近期提出的提取器构造，首次给出了与任意低次 $\mathbb{F}_2$-多项式源输出距离为 $1 - o(1)$ 的显式分布。此外，我们还描述了针对 $\mathsf{AC^0}[\oplus]$ 电路证明类似硬度结果的潜在路径。