  We present an alternating least squares type numerical optimization scheme to
estimate conditionally-independent mixture models in $\mathbb{R}^n$, without
parameterizing the distributions. Following the method of moments, we tackle an
incomplete tensor decomposition problem to learn the mixing weights and
componentwise means. Then we compute the cumulative distribution functions,
higher moments and other statistics of the component distributions through
linear solves. Crucially for computations in high dimensions, the steep costs
associated with high-order tensors are evaded, via the development of efficient
tensor-free operations. Numerical experiments demonstrate the competitive
performance of the algorithm, and its applicability to many models and
applications. Furthermore we provide theoretical analyses, establishing
identifiability from low-order moments of the mixture and guaranteeing local
linear convergence of the ALS algorithm.
