  In many applications, data come with a natural ordering. This ordering can
often induce local dependence among nearby variables. However, in complex data,
the width of this dependence may vary, making simple assumptions such as a
constant neighborhood size unrealistic. We propose a framework for learning
this local dependence based on estimating the inverse of the Cholesky factor of
the covariance matrix. Penalized maximum likelihood estimation of this matrix
yields a simple regression interpretation for local dependence in which
variables are predicted by their neighbors. Our proposed method involves
solving a convex, penalized Gaussian likelihood problem with a hierarchical
group lasso penalty. The problem decomposes into independent subproblems which
can be solved efficiently in parallel using first-order methods. Our method
yields a sparse, symmetric, positive definite estimator of the precision
matrix, encoding a Gaussian graphical model. We derive theoretical results not
found in existing methods attaining this structure. In particular, our
conditions for signed support recovery and estimation consistency rates in
multiple norms are as mild as those in a regression problem. Empirical results
show our method performing favorably compared to existing methods. We apply our
method to genomic data to flexibly model linkage disequilibrium. Our method is
also applied to improve the performance of discriminant analysis in sound
recording classification.
