Measurement Bounds for Compressed Sensing in Sensor Networks With Missing Data

In this paper, we study the problem of sparse vector recovery at the fusion center of a sensor network from linear sensor measurements when there is missing data. In the presence of missing data, the random sampling approach employed in compressed sensing provides excellent reconstruction accuracy. However, the theoretical guarantees associated with this sparse recovery problem have not been well studied. Therefore, in this paper, we derive a sufficient condition on the number of measurements required to ensure faithful recovery of a sparse signal using random (subGaussian) projections when the generation of missing data is modeled using a Bernoulli erasure channel. We analyze three different network topologies, namely, star, (relay aided-)tree, and serial-star topologies. Our analysis establishes how the minimum required number of measurements for recovery scales with the network parameters, the properties of the random measurement matrix, and the recovery algorithm. Finally, through numerical simulations, we study the minimum required number of measurements as a function of different system parameters and validate our theoretical results.