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 note that a similar bound was obtained concurrently and independently by Khodabandeh and Shinkar (ECCC '26). We also describe a potential avenue toward proving a similar hardness result for $\mathsf{AC^0}[\oplus]$ circuits.

翻译：我们提出了一种统一方法，用于构造针对受限计算模型难以生成的显式分布。我们的构造基于一种新的鲁棒提取器概念——即使少量数据点违反最小熵约束，这类提取器仍能保持可靠性。利用此类对象，我们证明：对于广泛的采样模型（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均为$1 - o(1)$，本质上恢复了以往所有已知的硬度结果。本工作扩展了Viola（SICOMP '14）的工作，他基于传统提取器构建了一个早期统一框架，用于排除极小误差的采样问题。作为我们技术的进一步应用，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。值得指出的是，Khodabandeh和Shinkar（ECCC '26）同时独立地获得了类似界值。我们还描述了证明$\mathsf{AC^0}[\oplus]$电路具有类似硬度结果的潜在途径。