Junta Correlation is Testable
Name
1904.04216.pdf
Description
Submitted version
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311.92 KB
Format
Adobe PDF
Checksum (MD5)
784d4d7914eba0942333693b56ad3378
Author(s)
De, Anindya
Mossel, Elchanan
Neeman, Joe
Date Issued
2019
Journal
Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
De, Anindya, Mossel, Elchanan and Neeman, Joe. 2019. "Junta Correlation is Testable." Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, 2019-November.
Version
Original manuscript
Abstract
© 2019 IEEE. The problem of tolerant junta testing is a natural and challenging problem which asks if the property of a function having some specified correlation with a k-Junta is testable. In this paper we give an affirmative answer to this question: There is an algorithm which given distance parameters c, d, and oracle access to a Boolean function f on the hypercube, has query complexity exp(k).poly(1/(c-d)) and distinguishes between the following cases: 1. The distance of f from any k-junta is at least c; 2. There is a k-junta g which has distance at most d from f. This is the first non-Trivial tester (i.e., query complexity is independent of the ambient dimension n) which works for all c and d (bounded by 0.5). The best previously known results by Blais et~al., required c to be at least 16d. In fact, with the same query complexity, we accomplish the stronger goal of identifying the most correlated k-junta, up to permutations of the coordinates. We can further improve the query complexity to poly(k/(c-d)) for the (weaker) task of distinguishing between the following cases: 1. The distance of f from any k'-junta is at least c. 2. There is a k-junta g which is at a distance at most d from f. Here k'=poly(k/(c-d)). Our main tools are Fourier analysis based algorithms that simulate oracle access to influential coordinates of functions.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Statistics and Data Science Center (Massachusetts Institute of Technology)
Massachusetts Institute of Technology. Institute for Data, Systems, and Society
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1109/FOCS.2019.00090
