Sparse Recovery with Graph Constraints: Fundamental Limits and Measurement Construction

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 $G$ ($M_{k,n}^G$). Our study suggests that $M_{k,n}^G$ may serve as a graph connectivity metric.