---
_id: '2815'
abstract:
- lang: eng
text: The fact that a sum of isotropic Gaussian kernels can have more modes than
kernels is surprising. Extra (ghost) modes do not exist in ℝ1 and are generally
not well studied in higher dimensions. We study a configuration of n+1 Gaussian
kernels for which there are exactly n+2 modes. We show that all modes lie on a
finite set of lines, which we call axes, and study the restriction of the Gaussian
mixture to these axes in order to discover that there are an exponential number
of critical points in this configuration. Although the existence of ghost modes
remained unknown due to the difficulty of finding examples in ℝ2, we show that
the resilience of ghost modes grows like the square root of the dimension. In
addition, we exhibit finite configurations of isotropic Gaussian kernels with
superlinearly many modes.
acknowledgement: This research is partially supported by the National Science Foundation
(NSF) under Grant DBI-0820624, by the European Science Foundation under the Research
Networking Programme, and the Russian Government Project 11.G34.31.0053.
article_processing_charge: No
article_type: original
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Brittany Terese
full_name: Fasy, Brittany Terese
id: F65D502E-E68D-11E9-9252-C644099818F6
last_name: Fasy
- first_name: Günter
full_name: Rote, Günter
last_name: Rote
citation:
ama: 'Edelsbrunner H, Fasy BT, Rote G. Add isotropic Gaussian kernels at own risk:
More and more resilient modes in higher dimensions. Discrete & Computational
Geometry. 2013;49(4):797-822. doi:10.1007/s00454-013-9517-x'
apa: 'Edelsbrunner, H., Fasy, B. T., & Rote, G. (2013). Add isotropic Gaussian
kernels at own risk: More and more resilient modes in higher dimensions. Discrete
& Computational Geometry. Springer. https://doi.org/10.1007/s00454-013-9517-x'
chicago: 'Edelsbrunner, Herbert, Brittany Terese Fasy, and Günter Rote. “Add Isotropic
Gaussian Kernels at Own Risk: More and More Resilient Modes in Higher Dimensions.”
Discrete & Computational Geometry. Springer, 2013. https://doi.org/10.1007/s00454-013-9517-x.'
ieee: 'H. Edelsbrunner, B. T. Fasy, and G. Rote, “Add isotropic Gaussian kernels
at own risk: More and more resilient modes in higher dimensions,” Discrete
& Computational Geometry, vol. 49, no. 4. Springer, pp. 797–822, 2013.'
ista: 'Edelsbrunner H, Fasy BT, Rote G. 2013. Add isotropic Gaussian kernels at
own risk: More and more resilient modes in higher dimensions. Discrete & Computational
Geometry. 49(4), 797–822.'
mla: 'Edelsbrunner, Herbert, et al. “Add Isotropic Gaussian Kernels at Own Risk:
More and More Resilient Modes in Higher Dimensions.” Discrete & Computational
Geometry, vol. 49, no. 4, Springer, 2013, pp. 797–822, doi:10.1007/s00454-013-9517-x.'
short: H. Edelsbrunner, B.T. Fasy, G. Rote, Discrete & Computational Geometry
49 (2013) 797–822.
date_created: 2018-12-11T11:59:44Z
date_published: 2013-06-01T00:00:00Z
date_updated: 2023-02-23T11:13:49Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s00454-013-9517-x
intvolume: ' 49'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s00454-013-9517-x
month: '06'
oa: 1
oa_version: Published Version
page: 797 - 822
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer
publist_id: '3991'
quality_controlled: '1'
related_material:
record:
- id: '3134'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: 'Add isotropic Gaussian kernels at own risk: More and more resilient modes
in higher dimensions'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 49
year: '2013'
...