  We are interested in the problem of learning the directed acyclic graph (DAG)
when data are generated from a linear structural equation model (SEM) and the
causal structure can be characterized by a polytree. Under the Gaussian
polytree models, we study sufficient conditions on the sample sizes for the
well-known Chow-Liu algorithm to exactly recover both the skeleton and the
equivalence class of the polytree, which is uniquely represented by a CPDAG. On
the other hand, necessary conditions on the required sample sizes for both
skeleton and CPDAG recovery are also derived in terms of information-theoretic
lower bounds, which match the respective sufficient conditions and thereby give
a sharp characterization of the difficulty of these tasks. We also consider the
problem of inverse correlation matrix estimation under the linear polytree
models, and establish the estimation error bound in terms of the dimension and
the total number of v-structures. We also consider an extension of group linear
polytree models, in which each node represents a group of variables. Our
theoretical findings are illustrated by comprehensive numerical simulations,
and experiments on benchmark data also demonstrate the robustness of polytree
learning when the true graphical structures can only be approximated by
polytrees.
