  Mixture modeling is a general technique for making any simple model more
expressive through weighted combination. This generality and simplicity in part
explains the success of the Expectation Maximization (EM) algorithm, in which
updates are easy to derive for a wide class of mixture models. However, the
likelihood of a mixture model is non-convex, so EM has no known global
convergence guarantees. Recently, method of moments approaches offer global
guarantees for some mixture models, but they do not extend easily to the range
of mixture models that exist. In this work, we present Polymom, an unifying
framework based on method of moments in which estimation procedures are easily
derivable, just as in EM. Polymom is applicable when the moments of a single
mixture component are polynomials of the parameters. Our key observation is
that the moments of the mixture model are a mixture of these polynomials, which
allows us to cast estimation as a Generalized Moment Problem. We solve its
relaxations using semidefinite optimization, and then extract parameters using
ideas from computer algebra. This framework allows us to draw insights and
apply tools from convex optimization, computer algebra and the theory of
moments to study problems in statistical estimation.
