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


Author(s): Edelsbrunner, Herbert; Fasy, Brittany T; Rote, Günter
Article Title: Add isotropic Gaussian kernels at own risk: More and more resilient modes in higher dimensions
Affiliation IST Austria
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.
Keywords: Diffusion; Critical points; Modes; Resilience; Convolution; Standard simplex; Unit Gaussian kernels
Journal Title: Discrete & Computational Geometry
Volume: 49
Issue 4
ISSN: 0179-5376
Publisher: Springer  
Date Published: 2013-06-01
Start Page: 797
End Page: 822
Sponsor: 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.
DOI: 10.1007/s00454-013-9517-x
Open access: no