  This work represents a natural coalescence of two important lines of work:
learning mixtures of Gaussians and algorithmic robust statistics. In particular
we give the first provably robust algorithm for learning mixtures of any
constant number of Gaussians. We require only mild assumptions on the mixing
weights (bounded fractionality) and that the total variation distance between
components is bounded away from zero. At the heart of our algorithm is a new
method for proving dimension-independent polynomial identifiability through
applying a carefully chosen sequence of differential operations to certain
generating functions that not only encode the parameters we would like to learn
but also the system of polynomial equations we would like to solve. We show how
the symbolic identities we derive can be directly used to analyze a natural
sum-of-squares relaxation.
