Blind Sparse Recovery Using Imperfect Sensor Networks

Abstract

This work investigates blind aggregation of structured highdimensional data, using a network of imperfect wireless sensor nodes which noncoherently communicate to a central fusion center or mobile data collector. In our setup, there is an unknown subset (of size ${k}$) of all $M$ registered autonomous transceiver nodes that sporadically wake up and simultaneously transmit their sensor readings through a shared channel. This procedure does particularly not involve a training phase that would allow for apriori channel predictions. In order to improve the resolvability in this noncoherent random access channel, the nodes perform an additional randomization of their signals. Since the transmission is usually imperfect, e.g., caused by low-quality hardware and unknown channel fading coefficients, the receiver measures a superposition of non-linearly distorted signals with unknown weights. Such a recovery task can be translated into a bilinear compressed sensing problem with rank-one measurements. We present a theoretical result for the Gaussian case which shows that $m = \mathcal {O}(sk\log (2nM/sk))$ measurements are sufficient to guarantee recovery of an $s$-sparse vector in $\mathbb {R}^{n}$. Moreover, our error bounds explicitly reflect the impact of the underlying non-linearities. The performance of our approach is also evaluated numerically for a random network generated by a compressible fading and node activity model.