At computer-vision group meeting this morning, Fadely, Foreman-Mackey, and I tri-coded Fadely's mixture of factor analyzers code and got it working. This is no mean feat, if you look at the linear algebra involved. And all that before noon!
The MFA model is like a mixture-of-Gaussians model, but each Gaussian has a variance tensor with reduced degrees of freedom: Each variance tensor is a sum of a diagonal (noise) tensor plus a low-rank general tensor. It is like a generalization of PCA to build a probabilistic distribution model for the data, then generalized to a mixture. It is a very powerful tool, because in large numbers of dimensions (for the data space) you get almost all the power of mixture-of-Gaussian modeling without the immense numbers of parameters.