  We provide two fundamental results on the population (infinite-sample)
likelihood function of Gaussian mixture models with $M \geq 3$ components. Our
first main result shows that the population likelihood function has bad local
maxima even in the special case of equally-weighted mixtures of well-separated
and spherical Gaussians. We prove that the log-likelihood value of these bad
local maxima can be arbitrarily worse than that of any global optimum, thereby
resolving an open question of Srebro (2007). Our second main result shows that
the EM algorithm (or a first-order variant of it) with random initialization
will converge to bad critical points with probability at least
$1-e^{-\Omega(M)}$. We further establish that a first-order variant of EM will
not converge to strict saddle points almost surely, indicating that the poor
performance of the first-order method can be attributed to the existence of bad
local maxima rather than bad saddle points. Overall, our results highlight the
necessity of careful initialization when using the EM algorithm in practice,
even when applied in highly favorable settings.
