Maximum Number of Modes of Gaussian Mixtures

Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density, or modes. In particular, the number of modes that can arise in a mixture of $k$ Gaussians in $d$ dimensions remains unknown in the general case. We give a brief account of this problem's history and expand on known results in this direction. In particular, we give improved lower bounds and the first upper bound on the maximum number of non-degenerate modes.