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Distribution testing lower bounds via reductions from communication complexity
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Blais, Eric, Canonne, Clément L. and Gur, Tom (2017) Distribution testing lower bounds via reductions from communication complexity. In: 32nd Computational Complexity Conference (CCC 2017). Published in: 32nd Computational Complexity Conference (CCC 2017), 79 28:1-28:40. ISBN 9783959770408. doi:10.4230/LIPIcs.CCC.2017.28 ISSN 1868-8969.
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Official URL: https://doi.org/10.4230/LIPIcs.CCC.2017.28
Abstract
We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef (Computational Complexity, 2012), we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds. Our main result is concerned with testing identity to a specific distribution p, given as a parameter. In a recent and influential work, Valiant and Valiant (FOCS, 2014) showed that the sample complexity of the aforementioned problem is closely related to the 2/3-quasinorm of p. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's K-functional. We show that this quantity is closely related to the size of the effective support of p (loosely speaking, the number of supported elements that constitute the vast majority of the mass of p). This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.
Item Type: | Conference Item (Paper) | ||||||
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Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||||
Library of Congress Subject Headings (LCSH): | Computational complexity, Distribution (Probability theory) | ||||||
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) | ||||||
Journal or Publication Title: | 32nd Computational Complexity Conference (CCC 2017) | ||||||
Publisher: | Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik | ||||||
Place of Publication: | Dagstuhl, Germany | ||||||
ISBN: | 9783959770408 | ||||||
ISSN: | 1868-8969 | ||||||
Official Date: | 21 July 2017 | ||||||
Dates: |
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Volume: | 79 | ||||||
Page Range: | 28:1-28:40 | ||||||
DOI: | 10.4230/LIPIcs.CCC.2017.28 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 11 April 2019 | ||||||
Date of first compliant Open Access: | 11 April 2019 | ||||||
Conference Paper Type: | Paper | ||||||
Title of Event: | 32nd Computational Complexity Conference (CCC 2017) | ||||||
Type of Event: | Conference | ||||||
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