  Approximate inference via information projection has been recently introduced
as a general-purpose approach for efficient probabilistic inference given
sparse variables. This manuscript goes beyond classical sparsity by proposing
efficient algorithms for approximate inference via information projection that
are applicable to any structure on the set of variables that admits enumeration
using a \emph{matroid}. We show that the resulting information projection can
be reduced to combinatorial submodular optimization subject to matroid
constraints. Further, leveraging recent advances in submodular optimization, we
provide an efficient greedy algorithm with strong optimization-theoretic
guarantees. The class of probabilistic models that can be expressed in this way
is quite broad and, as we show, includes group sparse regression, group sparse
principal components analysis and sparse canonical correlation analysis, among
others. Moreover, empirical results on simulated data and high dimensional
neuroimaging data highlight the superior performance of the information
projection approach as compared to established baselines for a range of
probabilistic models.
