On optimal sensor collaboration topologies for linear coherent estimation

Abstract —In the context of distributed estimation we considerthe problem of sensor collaboration, which refers to the actof sharing measurements with neighboring sensors prior totransmission to a fusion center. While incorporating the cost ofsensor collaboration, we aim to ﬁnd optimal sparse collaborationtopologies subject to a certain information or energy constraint.To achieve this goal, we present a tractable optimization frame-work and propose efﬁcient methods to solve the formulatedsensor collaboration problems. The effectiveness of our approachis demonstrated by numerical examples. I. I NTRODUCTION In this paper, we study the problem of distributed estimation[1], where each sensor reports the local observation of thephenomenon of interest and transmits a processed message(after inter-sensor communication) to a fusion center (FC)that determines the global estimate. The act of inter-sensorcommunication is referred to as sensor collaboration, wheresensors are allowed to share their observations with a set ofneighboring nodes prior to transmission to the FC. Recently,there has been a growing interest in designing sensor col-laboration schemes, spurred by a signiﬁcant improvement ofestimation performance resulting from collaboration [1]–[3].In [1], the optimal collaboration strategy was studied inan orthogonal multiple access channel (MAC) setting with afully connected network, where all the sensors are allowed tocollaborate. For a coherent MAC, it was shown in [2] thateven a partially connected network can yield performanceclose to that of a fully connected network; examples ofpartially connected networks include nearest-neighbor andrandom geometric graphs. However, the work of [1], [2]assumed that there is no cost associated with collaboration,the collaboration topologies are ﬁxed and given in advance,and the only unknowns are the collaboration weights used tocombine sensor observations. In [3], the nonzero collaborationcost was taken into account and a greedy algorithm wasdeveloped for seeking the optimal collaboration topology inenergy constrained sensor networks. This work is an extensionof [3]. We present a tractable framework to solve the collab-oration problem with nonzero collaboration cost, and proposean efﬁcient solution.First, we describe collaboration through the collaborationmatrix, in which the nonzero entries characterize the collabo-ration topology and the values of these entries characterizethe collaboration weights. We introduce a formulation thatsimultaneously optimizes both the collaboration topology andthe collaboration weights. This is in contrast to [3], wherethe optimization was performed in a sequential manner. Thenew formulation results in more efﬁcient allocation of energyresources as evidenced by improved distortion performance innumerical simulations.Second, in addition to the energy constrained problemconsidered in [3] where the Fisher information was maximizedsubject to a total energy budget, we also study an informationconstrained problem where the energy cost is minimizedwith a constraint on the Fisher information. For the informa-tion constrained problem, we apply the alternating directionsmethod of multipliers (ADMM) [4] to ﬁnd its locally optimalsolutions. For the energy constrained problem, we exploit itsrelationship with the information constrained problem andpropose a bisection algorithm to solve it. Numerical resultsare provided to illustrate the effectiveness of our proposedmethods.II. S