  We consider the statistical problem of recovering a hidden "ground truth"
binary labeling for the vertices of a graph up to low Hamming error from noisy
edge and vertex measurements. We present new algorithms and a sharp
finite-sample analysis for this problem on trees and sparse graphs with poor
expansion properties such as hypergrids and ring lattices. Our method
generalizes and improves over that of Globerson et al. (2015), who introduced
the problem for two-dimensional grid lattices.
  For trees we provide a simple, efficient, algorithm that infers the ground
truth with optimal Hamming error has optimal sample complexity and implies
recovery results for all connected graphs. Here, the presence of side
information is critical to obtain a non-trivial recovery rate. We then show how
to adapt this algorithm to tree decompositions of edge-subgraphs of certain
graph families such as lattices, resulting in optimal recovery error rates that
can be obtained efficiently
  The thrust of our analysis is to 1) use the tree decomposition along with
edge measurements to produce a small class of viable vertex labelings and 2)
apply an analysis influenced by statistical learning theory to show that we can
infer the ground truth from this class using vertex measurements. We show the
power of our method in several examples including hypergrids, ring lattices,
and the Newman-Watts model for small world graphs. For two-dimensional grids,
our results improve over Globerson et al. (2015) by obtaining optimal recovery
in the constant-height regime.
