Lower bounds for testing properties of functions over hypergrid domains

Eric Blais, Sofya Raskhodnikova, Grigory Yaroslavtsev

Research output: Chapter in Book/Report/Conference proceedingConference contribution

19 Scopus citations

Abstract

We show how the communication complexity method introduced in (Blais, Brody, Matulef 2012) can be used to prove lower bounds on the number of queries required to test properties of functions with non-hypercube domains. We use this method to prove strong, and in many cases optimal, lower bounds on the query complexity of testing fundamental properties of functions f : {1,⋯, n}d → R over hypergrid domains: monotonicity, the Lipschitz property, separate convexity, convexity and monotonicity of higher-order derivatives. There is a long line of work on upper bounds and lower bounds for many of these properties that uses a diverse set of combinatorial techniques. Our method provides a unified treatment of lower bounds for all these properties based on Fourier analysis. A key ingredient in our new lower bounds is a set of Walsh functions, a canonical Fourier basis for the set of functions on the line {1, ⋯ , n}. The orthogonality of the Walsh functions lets us use a product construction to extend our method from properties of functions over the line to properties of functions over hypergrids. Our product construction applies to properties over hypergrids that can be expressed in terms of axis-parallel directional derivatives, such as monotonicity, the Lipschitz property and separate convexity. We illustrate the robustness of our method by making it work for convexity, which is the property of the Hessian matrix of second derivatives being positive semidefinite and thus cannot be described by axisparallel directional derivatives alone. Such robustness contrasts with the state of the art in the upper bounds for testing properties over hypergrids: methods that work for other properties are not applicable for testing convexity, for which no nontrivial upper bounds are known for d ≥ 2..

Original languageEnglish (US)
Title of host publicationProceedings - IEEE 29th Conference on Computational Complexity, CCC 2014
PublisherIEEE Computer Society
Pages309-320
Number of pages12
ISBN (Print)9781479936267
DOIs
StatePublished - Jan 1 2014
Event29th Annual IEEE Conference on Computational Complexity, CCC 2014 - Vancouver, BC, Canada
Duration: Jun 11 2014Jun 13 2014

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Other

Other29th Annual IEEE Conference on Computational Complexity, CCC 2014
CountryCanada
CityVancouver, BC
Period6/11/146/13/14

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

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    Blais, E., Raskhodnikova, S., & Yaroslavtsev, G. (2014). Lower bounds for testing properties of functions over hypergrid domains. In Proceedings - IEEE 29th Conference on Computational Complexity, CCC 2014 (pp. 309-320). [6875499] (Proceedings of the Annual IEEE Conference on Computational Complexity). IEEE Computer Society. https://doi.org/10.1109/CCC.2014.38