Information Recovery from Pairwise Measurements

TL;DR

This paper develops an information-theoretic framework for recovering node-variables from noisy pairwise difference measurements represented by a graph, establishing fundamental conditions based on channel divergence and graph cuts.

Contribution

It introduces a unified approach using information theory to characterize exact recovery conditions for general graphs and measurement channels, extending to practical applications like stochastic block models and haplotype assembly.

Findings

Recovery depends on minimum channel divergence and graph cut size.

Order-wise tight conditions for exact recovery are derived for various models.

Sample complexity scales as n log n divided by a specific information measure.

Abstract

This paper is concerned with jointly recovering $n$ node-variables $\left\{ x_{i}\right\}_{1\leq i\leq n}$ from a collection of pairwise difference measurements. Imagine we acquire a few observations taking the form of $x_{i}-x_{j}$; the observation pattern is represented by a measurement graph $\mathcal{G}$ with an edge set $\mathcal{E}$ such that $x_{i}-x_{j}$ is observed if and only if $(i,j)\in\mathcal{E}$. To account for noisy measurements in a general manner, we model the data acquisition process by a set of channels with given input/output transition measures. Employing information-theoretic tools applied to channel decoding problems, we develop a \emph{unified} framework to characterize the fundamental recovery criterion, which accommodates general graph structures, alphabet sizes, and channel transition measures. In particular, our results isolate a family of \emph{minimum} \emph{channel divergence measures} to characterize the degree of measurement corruption, which together with the size of the minimum cut of $\mathcal{G}$ dictates the feasibility of exact information recovery. For various homogeneous graphs, the recovery condition depends almost only on the edge sparsity of the measurement graph irrespective of other graphical metrics; alternatively, the minimum sample complexity required for these graphs scales like \[ \text{minimum sample complexity }\asymp\frac{n\log n}{\mathsf{Hel}_{1/2}^{\min}} \] for certain information metric $\mathsf{Hel}_{1/2}^{\min}$ defined in the main text, as long as the alphabet size is not super-polynomial in $n$. We apply our general theory to three concrete applications, including the stochastic block model, the outlier model, and the haplotype assembly problem. Our theory leads to order-wise tight recovery conditions for all these scenarios.