  The Method of Moments [Pea94] is one of the most widely used methods in
statistics for parameter estimation, by means of solving the system of
equations that match the population and estimated moments. However, in practice
and especially for the important case of mixture models, one frequently needs
to contend with the difficulties of non-existence or non-uniqueness of
statistically meaningful solutions, as well as the high computational cost of
solving large polynomial systems. Moreover, theoretical analysis of the method
of moments are mainly confined to asymptotic normality style of results
established under strong assumptions.
  This paper considers estimating a $k$-component Gaussian location mixture
with a common (possibly unknown) variance parameter. To overcome the
aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise
the moment estimates by projecting to the truncated moment space (via
semidefinite programming) before solving the method of moments equations. Not
only does this regularization ensures existence and uniqueness of solutions, it
also yields fast solvers by means of Gauss quadrature. Furthermore, by proving
new moment comparison theorems in the Wasserstein distance via polynomial
interpolation and majorization techniques, we establish the statistical
guarantees and adaptive optimality of the proposed procedure, as well as oracle
inequality in misspecified models. These results can also be viewed as provable
algorithms for Generalized Method of Moments [Han82] which involves non-convex
optimization and lacks theoretical guarantees.
