Information Recovery from Pairwise Measurements: A Shannon-Theoretic Approach

This paper is concerned with jointly recovering n node-variables {x(1), ..., x(n)} from a collection of pairwise difference measurements. Specifically, several noisy measurements of x(i) - x(j) are acquired. This is represented by a graph with an edge set E such that x(i) - x(j) is observed only if (i, j) is an element of epsilon. To accommodate the noisy nature of data acquisition in a general way, we model the measurements by a set of channels with given input/output transition measures. Using information-theoretic tools applied to the channel decoding problem, we develop a unified framework to characterize a sufficient and a necessary condition for exact information recovery, which accommodates general graph structures, alphabet sizes, and channel transition measures. In particular, we isolate and highlight a family of minimum distance measures underlying the channel transition probabilities, which plays a central role in determining the recovery limits. For a broad class of homogeneous graphs, the recovery conditions we derive are tight up to some explicit constant, which depend only on the graph sparsity irrespective of other second-order graph metrics like the spectral gap.