Herbert Edelsbrunner, Brittany Terese Fasy, and Günter Rote:

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

  1. In: Proceedings of the 28th Annual Symposium on Computational Geometry, Chapel Hill, USA, June 17–20, 2012. Association for Computing Machinery, 2012, pp. 91–100. doi:10.1145/2261250.2261265free PDF @ACM
  2. Revised version, Discrete and Computational Geometry 49 (2013), 797–822. doi:10.1007/s00454-013-9517-x


The fact that the sum of isotropic Gaussian kernels (Gaussian distributions) can have more modes (local maxima) than kernels is surprising. In 2003, Carreira-Perpiñán and Williams exhibited n+1 isotropic Gaussian distributions in n dimensions with n+2 modes. Such extra (ghost) modes do not exist in one dimension and have not been well studied in higher dimensions.

We study a symmetric 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 the plane, 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.

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Last update: August 14, 2017.