Random Access Sensor Networks: Field Reconstruction From Incomplete Data Fatemeh Fazel Northeastern University Boston, MA Email: fazel@ece.neu.edu Maryam Fazel University of Washington Seattle, WA Email: mfazel@u.washington.edu Milica Stojanovic Northeastern University Boston, MA Email: millitsa@ece.neu.edu Abstract—We address efficient data gathering from a network of distributed sensors deployed in a challenging field environment with limited power and bandwidth. Utilizing the low-rank prop- erty of the sensing field, we leverage results from the matrix completion theory to integrate the sensing procedure with a simple and robust communication scheme based on random channel access. Results show that the space-time map of the sensing field can be recovered efficiently, using only a small subset of sensor measurements, collected over a fading random access channel. I. I NTRODUCTION Wireless sensor networks greatly facilitate long-term moni- toring of the natural environment [1]. Such networks comprise a large array of battery-powered sensors, a central collection unit referred to as the Fusion Center (FC) and a communica- tion scheme for the sensors to transmit their data to the FC. Once the network is deployed, access to the sensors is limited, hence re-charging the batteries is not easy. As a result, long- term deployment entails energy-aware sensing and efficient communication. Due to the size of the network, and the energy, bandwidth and time constraints of the data acquisition process, missing or partial information presents a ubiquitous problem in many applications such as geosciences [2], industrial and envi- ronmental monitoring, and monitoring surface deformations related to oil fields [3]. Thus, field recovery inevitably has to rely on incomplete data. Based on the principles of compressed sensing and ran- dom channel access, in [4], [5] we proposed an integrated architecture for sensing and communication, referred to as Random Access Compressed Sensing (RACS). This scheme targets the recovery of slowly varying fields from sub-sampled data, gathered while the process is approximately unchanged. RACS utilizes the fact that most natural signals have a sparse representation in the frequency domain. In the current work, we relax this assumption and take into consideration the temporal variations of the field over much longer periods of time. Employing results from the low- rank matrix recovery problem [6] and the matrix completion theory [7]–[9], we address the design of a random access network for long term monitoring of temporally varying fields. Research funded in part by ONR grant N00014-09-1-0700, NSF grant 0831728, and NSF CAREER grant ECCS-0847077. The matrix completion theory addresses the recovery of a low-rank matrix from a subset of its entries. Matrix completion is applied to field monitoring in [10], where the authors propose an online algorithm for subspace tracking. The process of gathering the data, which is the focus of our work, was not however in the scope of this paper. In the context of cognitive radios, collaborative spectrum sensing using matrix completion is studied in [11]. In [12], the authors discuss sampling strategies over a grid network to improve energy efficiency. The communication aspect, however, is not addressed, and the time variations of the process are not considered. The sensing and communication aspects of the wireless sensor networks are commonly treated independently. Our contribution is in unifying the sensing scheme, based on low- rank matrix recovery, with a simple and robust communication scheme using random access. The proposed integrated sensing and communication scheme relies only on a few assumptions about the statistical properties of the sensing field, and is thus applicable to a variety of fields. The rest of the paper is organized as follows: In Section II, we integrate the sensing scheme, based on matrix completion, with the communication scheme, using random access. In Section III, a probabilistic model for the system is provided upon which the network design guidelines of Section IV are then based. In Section V, we study the inherent trade-offs in choosing the system parameters. Finally, concluding remarks are provided in Section VI. II. RECONSTRUCTION FROM I NCOMPLETE DATA We consider a sensor network with N nodes, where each node measures the field at a given sensing rate λ 1 . We will rely on the matrix completion theory to determine the required sensing rate λ 1 , which is the objective of the design procedure. A. Field Model The network measures a process u(x, y, t) where x and y denote positions in space and t denotes time. We assume that node i is located at position (x i ,y i ), where i ∈{1,...,N }. The observations of the field are available at discrete intervals of time separated by Δt. Assuming a stationary process, the discretization interval Δt depends on the temporal autocor- relation of the process R(τ ), which quantifies the average correlation between two samples of the process separated