Information Recovery from Pairwise Measurements

by   Yuxin Chen, et al.

This paper is concerned with jointly recovering n node-variables { x_i}_1≤ i≤ 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 G with an edge set E such that x_i-x_j is observed if and only if (i,j)∈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 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 minimumchannel divergence measures to characterize the degree of measurement corruption, which together with the size of the minimum cut of 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 minimum sample complexity n n/Hel_1/2^ for certain information metric Hel_1/2^ 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.


page 1

page 2

page 3

page 4


Community Detection and Matrix Completion with Two-Sided Graph Side-Information

We consider the problem of recovering communities of users and communiti...

Graphical Model Inference with Erosely Measured Data

In this paper, we investigate the Gaussian graphical model inference pro...

Fundamental Limits on Data Acquisition: Trade-offs between Sample Complexity and Query Difficulty

In this paper, we consider query-based data acquisition and the correspo...

On the Difficulty of Selecting Ising Models with Approximate Recovery

In this paper, we consider the problem of estimating the underlying grap...

Graph Community Detection from Coarse Measurements: Recovery Conditions for the Coarsened Weighted Stochastic Block Model

We study the problem of community recovery from coarse measurements of a...

Sparse Signal Processing with Linear and Nonlinear Observations: A Unified Shannon-Theoretic Approach

We derive fundamental sample complexity bounds for recovering sparse and...

Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications

We consider a specific graph learning task: reconstructing a symmetric m...

Please sign up or login with your details

Forgot password? Click here to reset