  Learning the parameters of Gaussian mixture models is a fundamental and
widely studied problem with numerous applications. In this work, we give new
algorithms for learning the parameters of a high-dimensional, well separated,
Gaussian mixture model subject to the strong constraint of differential
privacy. In particular, we give a differentially private analogue of the
algorithm of Achlioptas and McSherry. Our algorithm has two key properties not
achieved by prior work: (1) The algorithm's sample complexity matches that of
the corresponding non-private algorithm up to lower order terms in a wide range
of parameters. (2) The algorithm does not require strong a priori bounds on the
parameters of the mixture components.
