Latent Variable Graphical Model Selection via Convex Optimization
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Venkat Chandrasekaran/CalTech

Suppose we have a Gaussian graphical model with sample observations of
only a subset of the variables. Can we separate the extra correlations
induced due to marginalization over the unobserved, hidden variables
from the structure among the observed variables? In other words is it
still possible to consistently perform model selection despite the
unobserved, latent variables? As we shall see the key problem that
arises is one of decomposing the concentration matrix of the observed
variables into a sparse matrix (representing graphical model structure
among the observed variables) and a low rank matrix (representing the
effects of marginalization over the hidden variables). Such a
decomposition can be accomplished by an estimator that is given by a
tractable convex program. This estimator performs consistent model
selection in the high-dimensional scaling regime in which the number
of observed/hidden variables grows with the number of samples of the
observed variables. The geometric aspects of our approach are
highlighted, with the algebraic varieties of sparse matrices and of
low-rank matrices playing an important role.

Joint work with Pablo Parrilo and Alan Willsky.