--- _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' ...