Filmus, Yuval ;
Leigh, Itai ;
Riazanov, Artur ;
Sokolov, Dmitry
Sampling and Certifying Symmetric Functions
Abstract
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 nbit strings of Hamming weight k denoted by Uⁿ_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 Uⁿ_k with an inversepolynomial error [Emanuele Viola, 2012; Czumaj, 2015]. We show that for every ε > 0 there exists τ such that for decision depth τ log (n/k) / log log (n/k), the error for sampling U_kⁿ is at least 1ε. Our result is based on the recent robust sunflower lemma [Ryan Alweiss et al., 2021; Rao, 2019].
Our second result is about matching a set of nbit strings with the image of a dlocal circuit, i.e. such that each output bit depends on at most d input bits. We study the set of all nbit strings whose Hamming weight is at least n/2. We improve the previously known locality lower bound from Ω(log^* n) [Beyersdorff et al., 2013] to Ω(√log n), leaving only a quartic gap from the best upper bound of O(log² n).
BibTeX  Entry
@InProceedings{filmus_et_al:LIPIcs.APPROX/RANDOM.2023.36,
author = {Filmus, Yuval and Leigh, Itai and Riazanov, Artur and Sokolov, Dmitry},
title = {{Sampling and Certifying Symmetric Functions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
pages = {36:136:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772969},
ISSN = {18688969},
year = {2023},
volume = {275},
editor = {Megow, Nicole and Smith, Adam},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18861},
URN = {urn:nbn:de:0030drops188611},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.36},
annote = {Keywords: sampling, lower bounds, robust sunflowers, decision trees, switching networks}
}
04.09.2023
Keywords: 

sampling, lower bounds, robust sunflowers, decision trees, switching networks 
Seminar: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Issue date: 

2023 
Date of publication: 

04.09.2023 