Sparse Recovery with Graph Constraints: Fundamental Limits and Measurement Construction

TL;DR

This paper investigates the fundamental limits and measurement strategies for sparse recovery in graphs, providing explicit constructions and bounds that relate to graph connectivity.

Contribution

It introduces a general measurement construction algorithm and derives order optimal bounds for measurements needed in graph-constrained sparse recovery.

Findings

Explicit measurement constructions for special graphs

Order optimal bounds for measurements in general graphs

Graph connectivity may serve as a metric for sparse recovery complexity

Abstract

This paper addresses the problem of sparse recovery with graph constraints in the sense that we can take additive measurements over nodes only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs. A general measurement construction algorithm is also proposed and evaluated. For any given graph $G$ with $n$ nodes, we derive order optimal upper bounds of the minimum number of measurements needed to recover any $k$-sparse vector over $G$ ($M^G_{k,n}$). Our study suggests that $M^G_{k,n}$ may serve as a graph connectivity metric.