  We present the multidimensional membership mixture (M3) models where every
dimension of the membership represents an independent mixture model and each
data point is generated from the selected mixture components jointly. This is
helpful when the data has a certain shared structure. For example, three unique
means and three unique variances can effectively form a Gaussian mixture model
with nine components, while requiring only six parameters to fully describe it.
In this paper, we present three instantiations of M3 models (together with the
learning and inference algorithms): infinite, finite, and hybrid, depending on
whether the number of mixtures is fixed or not. They are built upon Dirichlet
process mixture models, latent Dirichlet allocation, and a combination
respectively. We then consider two applications: topic modeling and learning 3D
object arrangements. Our experiments show that our M3 models achieve better
performance using fewer topics than many classic topic models. We also observe
that topics from the different dimensions of M3 models are meaningful and
orthogonal to each other.
