research-article

L_p-testing

Authors:

Sofya Raskhodnikova,

Grigory YaroslavtsevAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 164 - 173

https://doi.org/10.1145/2591796.2591887

Published: 31 May 2014 Publication History

Abstract

We initiate a systematic study of sublinear algorithms for approximately testing properties of real-valued data with respect to L_p distances for p = 1, 2. Such algorithms distinguish datasets which either have (or are close to having) a certain property from datasets which are far from having it with respect to L_p distance. For applications involving noisy real-valued data, using L_p distances allows algorithms to withstand noise of bounded L_p norm. While the classical property testing framework developed with respect to Hamming distance has been studied extensively, testing with respect to L_p distances has received little attention.

We use our framework to design simple and fast algorithms for classic problems, such as testing monotonicity, convexity and the Lipschitz property, and also distance approximation to monotonicity. In particular, for functions over the hypergrid domains [n]^d, the complexity of our algorithms for all these properties does not depend on the linear dimension n. This is impossible in the standard model. Most of our algorithms require minimal assumptions on the choice of sampled data: either uniform or easily samplable random queries suffice. We also show connections between the L_p-testing model and the standard framework of property testing with respect to Hamming distance. Some of our results improve existing bounds for Hamming distance.

Supplementary Material

MP4 File (p164-sidebyside.mp4)

Download
239.93 MB

References

[1]

N. Ailon, B. Chazelle, S. Comandur, and D. Liu. Estimating the distance to a monotone function. Random Struct. Algorithms, 31(3):371--383, 2007.

Digital Library

[2]

P. Awasthi, M. Jha, M. Molinaro, and S. Raskhodnikova. Testing Lipschitz functions on hypergrid domains. In APPROX-RANDOM, pages 387--398, 2012.

[3]

T. Batu, L. Fortnow, R. Rubinfeld, W. D. Smith, and P. White. Testing closeness of discrete distributions. J. ACM, 60(1):4, 2013.

Digital Library

[4]

E. Blais, S. Raskhodnikova, and G. Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In CCC, 2014.

Digital Library

[5]

D. Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and Lipschitz testing over hypercubes and hypergrids. In STOC, pages 419--428, 2013.

Digital Library

[6]

D. Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. In APPROXRANDOM, pages 425--435, 2013.

[7]

D. Chakrabarty, K. Dixit, M. Jha, and C. Seshadhri. Optimal lower bounds for Lipschitz testing via monotonicity. Private communication, 2013.

[8]

K. Dixit, M. Jha, S. Raskhodnikova, and A. Thakurta. Testing the Lipschitz property over product distributions with applications to data privacy. In TCC, pages 418--436, 2013.

Digital Library

[9]

Y. Dodis, O. Goldreich, E. Lehman, S. Raskhodnikova, D. Ron, and A. Samorodnitsky. Improved testing algorithms for monotonicity. In RANDOM, pages 97--108, 1999.

Digital Library

[10]

C. Dwork, F. McSherry, K. Nissim, and A. Smith. Calibrating noise to sensitivity in private data analysis. In TCC, pages 265--284, 2006.

Digital Library

[11]

F. Ergun, S. Kannan, S. R. Kumar, R. Rubinfeld, and M. Viswanathan. Spot-checkers. J. Comput. Syst. Sci., 60 (3):717--751, 2000.

Digital Library

[12]

S. Fattal and D. Ron. Approximating the distance to convexity. Available at: http://www.eng.tau.ac.il/~danar/Public-pdf/app-conv. pdf, 2007.

[13]

S. Fattal and D. Ron. Approximating the distance to monotonicity in high dimensions. ACM Transactions on Algorithms, 6(3), 2010.

Digital Library

[14]

V. Feldman and J. Vondrák. Optimal bounds on approximation of submodular and XOS functions by juntas. In FOCS, pages 227--236, 2013.

Digital Library

[15]

E. Fischer. On the strength of comparisons in property testing. Inf. Comput., 189(1):107--116, 2004.

Digital Library

[16]

O. Goldreich. On multiple input problems in property testing. Electronic Colloquium on Computational Complexity (ECCC), 20:67, 2013.

[17]

O. Goldreich and D. Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302--343, 2002.

Digital Library

[18]

O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653--750, 1998.

Digital Library

[19]

O. Goldreich, S. Goldwasser, E. Lehman, D. Ron, and A. Samorodnitsky. Testing monotonicity. Combinatorica, 20(3):301--337, 2000.

[20]

D. Haussler. Decision theoretic generalizations of the PAC model for neural net and other learning applications. Inf. Comput., 100(1):78--150, 1992.

Digital Library

[21]

M. Jha and S. Raskhodnikova. Testing and reconstruction of Lipschitz functions with applications to data privacy. SIAM J. Comput., 42(2):700--731, 2013.

[22]

L. A. Levin. One-way functions and pseudorandom generators. In STOC, pages 363--365, 1985.

Digital Library

[23]

M. Parnas, D. Ron, and R. Rubinfeld. Tolerant property testing and distance approximation. J. Comput. Syst. Sci., 72(6):1012--1042, 2006.

Digital Library

[24]

L. Rademacher and S. Vempala. Testing geometric convexity. In FSTTCS, pages 469--480, 2004.

Digital Library

[25]

S. Raskhodnikova. Approximate testing of visual properties. In RANDOM-APPROX, pages 370--381, 2003.

[26]

R. Rubinfeld and M. Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252--271, 1996.

Digital Library

[27]

M. Saks and C. Seshadhri. Estimating the longest increasing sequence in polylogarithmic time. In FOCS, pages 458--467, 2010.

Digital Library

Cited By

Berman PMurzabulatov MRaskhodnikova SRistache D(2024)Testing Connectedness of ImagesAlgorithmica10.1007/s00453-024-01248-xOnline publication date: 12-Sep-2024
https://doi.org/10.1007/s00453-024-01248-x
Levi APallavoor RRaskhodnikova SVarma N(2021)Erasure-Resilient Sublinear-Time Graph AlgorithmsACM Transactions on Computation Theory10.1145/348825014:1(1-22)Online publication date: 15-Dec-2021
https://dl.acm.org/doi/10.1145/3488250
Chen XDe AServedio RMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)Testing noisy linear functions for sparsityProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384239(610-623)Online publication date: 22-Jun-2020
https://dl.acm.org/doi/10.1145/3357713.3384239
Show More Cited By

Index Terms

L_p-testing
1. Theory of computation
  1. Design and analysis of algorithms
  2. Models of computation

Recommendations

Property Testing on Product Distributions: Optimal Testers for Bounded Derivative Properties
Special Issue on SODA'15 and Regular Papers

The primary problem in property testing is to decide whether a given function satisfies a certain property or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the ...
Testing linear-invariant non-linear properties: a short report
Property testing

The rich collection of successes in property testing raises a natural question: Why are so many different properties turning out to be locally testable? Are there some broad "features" of properties that make them testable? Kaufman and Sudan (STOC 2008) ...
Tolerant property testing and distance approximation

In this paper we study a generalization of standard property testing where the algorithms are required to be more tolerant with respect to objects that do not have, but are close to having, the property. Specifically, a tolerant property testing ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 31 May 2014

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Division of Computing and Communication Foundations
College of Engineering Fellowship at Penn State
Hariri Institute for Computing and Computational Science and Engineering at Boston University
Brown University

Conference

STOC '14

Sponsor:

SIGACT

STOC '14: Symposium on Theory of Computing

May 31 - June 3, 2014

New York, New York

Acceptance Rates

STOC '14 Paper Acceptance Rate 91 of 319 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

16
Total Citations
View Citations
273
Total Downloads

Downloads (Last 12 months)16
Downloads (Last 6 weeks)3

Reflects downloads up to 04 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Berman PMurzabulatov MRaskhodnikova SRistache D(2024)Testing Connectedness of ImagesAlgorithmica10.1007/s00453-024-01248-xOnline publication date: 12-Sep-2024
https://doi.org/10.1007/s00453-024-01248-x
Levi APallavoor RRaskhodnikova SVarma N(2021)Erasure-Resilient Sublinear-Time Graph AlgorithmsACM Transactions on Computation Theory10.1145/348825014:1(1-22)Online publication date: 15-Dec-2021
https://dl.acm.org/doi/10.1145/3488250
Chen XDe AServedio RMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)Testing noisy linear functions for sparsityProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384239(610-623)Online publication date: 22-Jun-2020
https://dl.acm.org/doi/10.1145/3357713.3384239
Ben-Eliezer OCanonne CLetzter SWaingarten E(2019)Finding Monotone Patterns in Sublinear Time2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2019.000-1(1469-1494)Online publication date: Nov-2019
https://doi.org/10.1109/FOCS.2019.000-1
Berman PMurzabulatov MRaskhodnikova S(2019)The Power and Limitations of Uniform Samples in Testing Properties of FiguresAlgorithmica10.1007/s00453-018-0467-981:3(1247-1266)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1007/s00453-018-0467-9
Ben-Eliezer OFischer EServedio R(2018)Earthmover resilience and testing in ordered structuresProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235604(1-35)Online publication date: 22-Jun-2018
https://dl.acm.org/doi/10.5555/3235586.3235604
Black HChakrabarty DSeshadhri CCzumaj A(2018)A o(d) · polylog n monotonicity tester for boolean functions over the hypergrid [n]Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175444(2133-2151)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175444
Georgakopoulos ATyros K(2018) The Bradley–Terry condition is -testable Discrete Mathematics10.1016/j.disc.2017.10.026341:4(1171-1177)Online publication date: Apr-2018
https://doi.org/10.1016/j.disc.2017.10.026
Berman PMurzabulatov MRaskhodnikova S(2018)Testing convexity of figures under the uniform distributionRandom Structures & Algorithms10.1002/rsa.2079754:3(413-443)Online publication date: 19-Sep-2018
https://doi.org/10.1002/rsa.20797
Pallavoor RRaskhodnikova SVarma N(2017)Parameterized Property Testing of FunctionsACM Transactions on Computation Theory10.1145/31552969:4(1-19)Online publication date: 14-Dec-2017
https://dl.acm.org/doi/10.1145/3155296
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents