Hard-to-Sample Distributions from Robust Extractors
For theoretical computer scientists studying pseudorandomness and circuit complexity, this provides a unified framework for proving sampling hardness, with a new result for low-degree polynomial sources.
The paper introduces robust extractors to construct explicit distributions that are hard for various restricted computational models to sample, achieving distance 1-o(1) from any output of the model. It recovers previous hardness results and provides the first explicit distribution with such distance from low-degree F2-polynomial sources.
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.