Localization from connectivity: 1-Bit maximum likelihood approach

We consider the problem of determining the location of sensor nodes in a wireless sensor ad hoc network when only connectivity information is available, i.e., one only knows if a pair of nodes are within a fixed radio range (distance) of each other but does not have access to exact or even approximate distance information. We propose a maximum likelihood based reconstruction algorithm to reconstruct the node positions in a d-dimensional Euclidean space. The original problem formulation results in constrained minimization of a negative log-likelihood with respect to n × n symmetric positive semidefinite matrices Q for an n-node network, and is a convex optimization problem. We recast this problem into a non-convex optimization problem with respect to X where Q = X ^T X. We link our algorithm to existing results which confirm that the nonconvex optimization problem with m × n X provides a global solution to its convex equivalent for certain choices of m. We demonstrate that our algorithm is empirically successful for both uniform and irregular networks, using only a few anchor nodes. Finally, numerical experiments demonstrate improved performance of the proposed algorithm relative to the MDS algorithm and a variant.