Inference in Sparse Graphs with Pairwise Measurements and Side Information

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.