research-article

Add isotropic Gaussian kernels at own risk: more and more resilient modes in higher dimensions

Authors:

Herbert Edelsbrunner,

Brittany Terese Fasy,

Günter RoteAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 91 - 100

https://doi.org/10.1145/2261250.2261265

Published: 17 June 2012 Publication History

Abstract

It has been an open question whether the sum of finitely many isotropic Gaussian kernels in n ≥ 2 dimensions can have more modes than kernels, until in 2003 Carreira-Perpinan and Williams exhibited n+1 isotropic Gaussian kernels in Rⁿ with n+2 modes. We give a detailed analysis of this example, showing that it has exponentially many critical points and that the resilience of the extra mode grows like √n. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.

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Cited By

Zheng YPhillips JCao LZhang CJoachims TWebb GMargineantu DWilliams G(2015)L∞ Error and Bandwidth Selection for Kernel Density Estimates of Large DataProceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining10.1145/2783258.2783357(1533-1542)Online publication date: 10-Aug-2015
https://dl.acm.org/doi/10.1145/2783258.2783357
Genovese CPerone-Pacifico MVerdinelli IWasserman L(2014)Nonparametric ridge estimationThe Annals of Statistics10.1214/14-AOS121842:4Online publication date: 1-Aug-2014
https://doi.org/10.1214/14-AOS1218
Agarwal PHar-Peled SKaplan HSharir M(2014)Union of Random Minkowski Sums and Network Vulnerability AnalysisDiscrete & Computational Geometry10.1007/s00454-014-9626-152:3(551-582)Online publication date: 1-Oct-2014
https://dl.acm.org/doi/10.1007/s00454-014-9626-1
Show More Cited By

Index Terms

Add isotropic Gaussian kernels at own risk: more and more resilient modes in higher dimensions
1. Computing methodologies
  1. Artificial intelligence
    1. Computer vision
      1. Computer vision representations
        Shape representations
2. Mathematics of computing
  1. Probability and statistics
    1. Distribution functions

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cover image ACM Conferences

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

Copyright © 2012 ACM.

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Publication History

Published: 17 June 2012

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SoCG '12

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SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Zheng YPhillips JCao LZhang CJoachims TWebb GMargineantu DWilliams G(2015)L∞ Error and Bandwidth Selection for Kernel Density Estimates of Large DataProceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining10.1145/2783258.2783357(1533-1542)Online publication date: 10-Aug-2015
https://dl.acm.org/doi/10.1145/2783258.2783357
Genovese CPerone-Pacifico MVerdinelli IWasserman L(2014)Nonparametric ridge estimationThe Annals of Statistics10.1214/14-AOS121842:4Online publication date: 1-Aug-2014
https://doi.org/10.1214/14-AOS1218
Agarwal PHar-Peled SKaplan HSharir M(2014)Union of Random Minkowski Sums and Network Vulnerability AnalysisDiscrete & Computational Geometry10.1007/s00454-014-9626-152:3(551-582)Online publication date: 1-Oct-2014
https://dl.acm.org/doi/10.1007/s00454-014-9626-1
Agarwal PKaplan HSharir Mda Fonseca GLewiner TPeñaranda LChan TKlein R(2013)Union of random minkowski sums and network vulnerability analysisProceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462398(177-186)Online publication date: 17-Jun-2013
https://dl.acm.org/doi/10.1145/2462356.2462398

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