  Modeling the distribution of high dimensional data by a latent tree graphical
model is a prevalent approach in multiple scientific domains. A common task is
to infer the underlying tree structure, given only observations of its terminal
nodes. Many algorithms for tree recovery are computationally intensive, which
limits their applicability to trees of moderate size. For large trees, a common
approach, termed divide-and-conquer, is to recover the tree structure in two
steps. First, recover the structure separately of multiple, possibly random
subsets of the terminal nodes. Second, merge the resulting subtrees to form a
full tree. Here, we develop Spectral Top-Down Recovery (STDR), a deterministic
divide-and-conquer approach to infer large latent tree models. Unlike previous
methods, STDR partitions the terminal nodes in a non random way, based on the
Fiedler vector of a suitable Laplacian matrix related to the observed nodes. We
prove that under certain conditions, this partitioning is consistent with the
tree structure. This, in turn, leads to a significantly simpler merging
procedure of the small subtrees. We prove that STDR is statistically consistent
and bound the number of samples required to accurately recover the tree with
high probability. Using simulated data from several common tree models in
phylogenetics, we demonstrate that STDR has a significant advantage in terms of
runtime, with improved or similar accuracy.
