Sparse Recovery with Graph Constraints

TL;DR

This paper develops measurement strategies for sparse signal recovery constrained by network topology, providing explicit constructions and bounds that outperform random methods, especially for line networks, and analyzing the impact of graph structure.

Contribution

It introduces explicit measurement constructions for graph-constrained sparse recovery and establishes bounds, including an optimal method for line networks and a general algorithm for arbitrary graphs.

Findings

Explicit measurement constructions for special graphs

Number of measurements less than existing random methods

Optimal measurement construction for line networks

Abstract

Sparse recovery can recover sparse signals from a set of underdetermined linear measurements. Motivated by the need to monitor large-scale networks from a limited number of measurements, this paper addresses the problem of recovering sparse signals in the presence of network topological constraints. Unlike conventional sparse recovery where a measurement can contain any subset of the unknown variables, we use a graph to characterize the topological constraints and allow an additive measurement over nodes (unknown variables) only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs, and the number of measurements by our construction is less than that needed by existing random constructions. Moreover, our construction for a line network is provably optimal in the sense that it requires the minimum number of measurements. A measurement construction algorithm for general graphs is also proposed and evaluated. For any given graph $G$ with $n$ nodes, we derive bounds of the minimum number of measurements needed to recover any $k$-sparse vector over $G$ ($M^G_{k,n}$). Using the Erd\H{o}s-R\'enyi random graph as an example, we characterize the dependence of $M^G_{k,n}$ on the graph structure.