Information Recovery from Pairwise Measurements

TL;DR

This paper analyzes the fundamental limits of recovering objects' values from noisy, incomplete pairwise measurements over various graph models, and proposes algorithms to approach these limits.

Contribution

It characterizes the minimax recovery rate under a general outlier model and develops algorithms for near-optimal recovery depending on the group size and graph sparsity.

Findings

Recovery rate depends mainly on graph sparsity.

Proposed algorithms approach the theoretical limit for large group sizes.

Fundamental limits decay with group size before connectivity constraints dominate.

Abstract

A variety of information processing tasks in practice involve recovering $n$ objects from single-shot graph-based measurements, particularly those taken over the edges of some measurement graph $\mathcal{G}$. This paper concerns the situation where each object takes value over a group of $M$ different values, and where one is interested to recover all these values based on observations of certain pairwise relations over $\mathcal{G}$. The imperfection of measurements presents two major challenges for information recovery: 1) $\textit{inaccuracy}$: a (dominant) portion $1-p$ of measurements are corrupted; 2) $\textit{incompleteness}$: a significant fraction of pairs are unobservable, i.e. $\mathcal{G}$ can be highly sparse. Under a natural random outlier model, we characterize the $\textit{minimax recovery rate}$, that is, the critical threshold of non-corruption rate $p$ below which exact information recovery is infeasible. This accommodates a very general class of pairwise relations. For various homogeneous random graph models (e.g. Erdos Renyi random graphs, random geometric graphs, small world graphs), the minimax recovery rate depends almost exclusively on the edge sparsity of the measurement graph $\mathcal{G}$ irrespective of other graphical metrics. This fundamental limit decays with the group size $M$ at a square root rate before entering a connectivity-limited regime. Under the Erdos Renyi random graph, a tractable combinatorial algorithm is proposed to approach the limit for large $M$ ($M=n^{\Omega(1)}$), while order-optimal recovery is enabled by semidefinite programs in the small $M$ regime. The extended (and most updated) version of this work can be found at (http://arxiv.org/abs/1504.01369).