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

H. Edelsbrunner, B. Fasy, G. Rote, Discrete & Computational Geometry 49 (2013) 797–822.

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Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2013-06-01
Journal Title
Discrete & Computational Geometry
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.
Volume
49
Issue
4
Page
797 - 822
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Edelsbrunner H, Fasy B, 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
Edelsbrunner, H., Fasy, B., & 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. https://doi.org/10.1007/s00454-013-9517-x
Edelsbrunner, Herbert, Brittany Fasy, and Günter Rote. “Add Isotropic Gaussian Kernels at Own Risk: More and More Resilient Modes in Higher Dimensions.” Discrete & Computational Geometry 49, no. 4 (2013): 797–822. https://doi.org/10.1007/s00454-013-9517-x.
H. Edelsbrunner, B. 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, pp. 797–822, 2013.
Edelsbrunner H, Fasy B, 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.
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.

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