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

Authors:
Herbert Edelsbrunner;Brittany Terese Fasy;Günter Rote
Affiliations:
IST Austria, Duke University, Yaroslavl' State University, Geomagic, Klosterneuburg, Austria;Duke University, IST Austria, Durham, NC, USA;Freie Universität Berlin, Berlin, Germany
Venue:
Proceedings of the twenty-eighth annual symposium on Computational geometry
Year:
2012

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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.