Add Isotropic Gaussian Kernels at Own Risk: More and More Resilient Modes in Higher Dimensions

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 $${{\mathbb{R }}}^1$$R1 and are generally not well studied in higher dimensions. We study a configuration of $$n+1$$n+1 Gaussian kernels for which there are exactly $$n+2$$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 $${{\mathbb{R }}}^2$$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.