Sampling and Certifying Symmetric Functions
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A circuit 𝒞 samples a distribution 𝐗 with an error ϵ if the statistical distance between the output of 𝒞 on the uniform input and 𝐗 is ϵ. We study the hardness of sampling a uniform distribution over the set of n-bit strings of Hamming weight k denoted by 𝐔^n_k for _decision forests_, i.e. every output bit is computed as a decision tree of the inputs. For every k there is an O(log n)-depth decision forest sampling 𝐔^n_k with an inverse-polynomial error [Viola 2012, Czumaj 2015]. We show that for every ϵ > 0 there exists τ such that for decision depth τlog (n/k) / loglog (n/k), the error for sampling 𝐔_k^n is at least 1-ϵ. Our result is based on the recent robust sunflower lemma [Alweiss, Lovett, Wu, Zhang 2021, Rao 2019]. Our second result is about matching a set of n-bit strings with the image of a d-_local_ circuit, i.e. such that each output bit depends on at most d input bits. We study the set of all n-bit strings whose Hamming weight is at least n/2. We improve the previously known locality lower bound from Ω(log^* n) [Beyersdorff, Datta, Krebs, Mahajan, Scharfenberger-Fabian, Sreenivasaiah, Thomas and Vollmer, 2013] to Ω(√(log n)), leaving only a quartic gap from the best upper bound of O(log^2 n).
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