Herbert Edelsbrunner, Brittany Terese Fasy, and Günter Rote:
Add isotropic Gaussian kernels at own risk: more and more resilient modes
in higher dimensions

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:10.1145/2261250.2261265, free PDF @ACM

Revised version,
Discrete and Computational
Geometry 49 (2013), 797–822.
doi:10.1007/s004540139517x
Abstract
The fact that the sum of isotropic Gaussian kernels (Gaussian distributions)
can have more modes (local maxima) than kernels is surprising. In 2003,
CarreiraPerpiñán 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.
Last update: August 14, 2017.