🤖 AI Summary
This work addresses the problem of explicitly constructing distributions that are nearly unsamplable by restricted computational models, such as low-depth circuits or small-space sources. To this end, we introduce a novel robust extractor that remains effective even when a small number of samples fail to satisfy the standard min-entropy assumption. Leveraging this tool, we unify and generalize existing hardness results, yielding the first explicit distribution that is statistically far from any source samplable by low-degree 𝔽₂ polynomials—specifically, at statistical distance 1−o(1) from the outputs of various restricted sampling models. Our approach also provides a new route toward establishing sampling hardness for AC⁰[⊕] circuits.
📝 Abstract
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.