research-article

The average sensitivity of an intersection of half spaces

Author:

Daniel M. KaneAuthors Info & Claims

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

Pages 437 - 440

https://doi.org/10.1145/2591796.2591798

Published: 31 May 2014 Publication History

Get Access

Abstract

We prove new bounds on the average sensitivity of the indicator function of an intersection of k halfspaces. In particular, we prove the optimal bound of O(√nlog(k)). This generalizes a result of Nazarov, who proved the analogous result in the Gaussian case, and improves upon a result of Harsha, Klivans and Meka. Furthermore, our result has implications for the runtime required to learn intersections of halfspaces.

Supplementary Material

MP4 File (p437-sidebyside.mp4)

Download
239.65 MB

References

[1]

I. Diakonikolas, P. Harsha, A. Klivans, R. Meka, P. Raghavendra, R. A. Servedio, and L.-Y. Tan. Bounding the average sensitivity and noise sensitivity of polynomial threshold functions. Proceedings of the 42nd ACM symposium on Theory of Computing, 2010.

Digital Library

Google Scholar

[2]

I. Diakonikolas, P. Raghavendra, R. A. Servedio, and L.-Y. Tan. Average sensitivity and noise sensitivity of polynomial threshold functions. manuscript. available at http://arxiv.org/abs/0909.5011.

Google Scholar

[3]

C. Gotsman and N. Linial. Spectral properties of threshold functions. Combinatorica, 14(1):35âĂŞ--50, 1994.

Google Scholar

[4]

P. Harsha, A. R. Klivans, and R. Meka. An invariance principle for polytopes. Symposium on Theory of Computing, 2010.

Digital Library

Google Scholar

[5]

A. Kalai, A. R. Klivans, Y. Mansour, and R. Servedio. Agnostically learning halfspaces. Foundations of Computer Science, 2005.

Digital Library

Google Scholar

[6]

D. M. Kane. The gaussian surface area and noise sensitivity of degree-d polynomial threshold functions. Proceedings of Conference on Computational Complexity, pages 205--210, 2010.

Digital Library

Google Scholar

[7]

D. M. Kane. The correct exponent for the gotsman-linial conjecture. Conference on Computational Complexity, 2013.

Crossref

Google Scholar

[8]

A. R. Klivans, R. O'Donnell, and R. A. Servedio. Learning geometric concepts via gaussian surface area. Proceedings of the 49th Foundations of Computer Science, 49:541--550, 2008.

Digital Library

Google Scholar

Cited By

View all

O’Donnell RServedio RTan L(2022)Fooling PolytopesJournal of the ACM10.1145/346053269:2(1-37)Online publication date: 31-Jan-2022
https://dl.acm.org/doi/10.1145/3460532
O'Donnell RServedio RTan LCharikar MCohen E(2019)Fooling polytopesProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316321(614-625)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316321
Diakonikolas IKane DStewart ADiakonikolas IKempe DHenzinger M(2018)Learning geometric concepts with nasty noiseProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188754(1061-1073)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188754
Show More Cited By

Index Terms

The average sensitivity of an intersection of half spaces
1. Computing methodologies
  1. Machine learning
2. Mathematics of computing
  1. Discrete mathematics

Recommendations

Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-d polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube {-1,1}ⁿ and for PTFs over Rⁿ under the standard n-...
Hardness amplification within NP
Special issue on computational complexity 2002

In this paper we investigate the following question: if NP is slightly hard on average, is it very hard on average? We give a positive answer: if there is a function in NP which is infinitely often balanced and (1 - 1/poly(n))-hard for circuits of ...
The complexity of constructing pseudorandom generators from hard functions

We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constant-depth circuits. We show that, starting from a function f : {0,1}^l --> {0,1} computable in alternating time O(l) with O(1) alternations that ...

Comments

Information & Contributors

Information

Published In

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

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 the author(s) 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].

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

National Science Foundation

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%

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
123
Total Downloads

Downloads (Last 12 months)5
Downloads (Last 6 weeks)0

Reflects downloads up to 26 Jul 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

O’Donnell RServedio RTan L(2022)Fooling PolytopesJournal of the ACM10.1145/346053269:2(1-37)Online publication date: 31-Jan-2022
https://dl.acm.org/doi/10.1145/3460532
O'Donnell RServedio RTan LCharikar MCohen E(2019)Fooling polytopesProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316321(614-625)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316321
Diakonikolas IKane DStewart ADiakonikolas IKempe DHenzinger M(2018)Learning geometric concepts with nasty noiseProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188754(1061-1073)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188754
Servedio RTan L(2017)Fooling Intersections of Low-Weight Halfspaces2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.81(824-835)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.81

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.

Cited By

Index Terms

Recommendations

Bounding the average sensitivity and noise sensitivity of polynomial threshold functions

Hardness amplification within NP

The complexity of constructing pseudorandom generators from hard functions

Comments

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Author Tags

Qualifiers

Funding Sources

Conference

Acceptance Rates

Other Metrics

Article Metrics

Other Metrics

Cited By

Login options

Full Access

PDF

eReader

Abstract

Supplementary Material

References

Cited By

Index Terms

Recommendations

Bounding the average sensitivity and noise sensitivity of polynomial threshold functions

Hardness amplification within NP

The complexity of constructing pseudorandom generators from hard functions

Comments

Information

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Author Tags

Qualifiers

Funding Sources

Conference

Acceptance Rates

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

Get Access

Login options

Full Access

View options

PDF

eReader

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations