Low-degree tests at large distances

Alex Samorodnitsky*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

87 Scopus citations

Abstract

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. A central step in our analysis of quadraticity tests is the proof of aninverse theorem for the third Gowers uniformity norm of boolean functions. The last result implies that it ispossible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius. Our main technical tools are Fourier analysis on Z 2 n and methods from additive number theory. We observe that these methods can be used to give a tight analysis of the Abelian Homomorphism testing problemfor some families of groups, including powers of Z p.

Original languageAmerican English
Title of host publicationSTOC'07
Subtitle of host publicationProceedings of the 39th Annual ACM Symposium on Theory of Computing
Pages506-515
Number of pages10
DOIs
StatePublished - 2007
EventSTOC'07: 39th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: 11 Jun 200713 Jun 2007

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

ConferenceSTOC'07: 39th Annual ACM Symposium on Theory of Computing
Country/TerritoryUnited States
CitySan Diego, CA
Period11/06/0713/06/07

Bibliographical note

Funding Information:
We would like to thank Jay Anderson for providing us with his suite of PSF-fitting and image alignment software, and for his valuable instruction, guidance and technical support. We also thank the anonymous referee for a constructive review. Support for this work was provided by NASA grants GO-13390 and GO-13791 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute. This work has made use of data from the European Space Agency (ESA) mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

Keywords

  • Low-degree tests

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