  We investigate the landscape of the negative log-likelihood function of
Gaussian Mixture Models (GMMs) with a general number of components in the
population limit. As the objective function is non-convex, there can be
multiple local minima that are not globally optimal, even for well-separated
mixture models. Our study reveals that all local minima share a common
structure that partially identifies the cluster centers (i.e., means of the
Gaussian components) of the true location mixture. Specifically, each local
minimum can be represented as a non-overlapping combination of two types of
sub-configurations: fitting a single mean estimate to multiple Gaussian
components or fitting multiple estimates to a single true component. These
results apply to settings where the true mixture components satisfy a certain
separation condition, and are valid even when the number of components is over-
or under-specified. We also present a more fine-grained analysis for the
setting of one-dimensional GMMs with three components, which provide sharper
approximation error bounds with improved dependence on the separation.
