The distributed Gauss-Newton methods for solving the inverse of approximated Hessian with application to target localization

Gauss-Newton method is a Hessian free second-order Newton's optimization method, which can be used to solve the common nonlinear least squares (NLLS) problems. In this paper, we consider the collaborative target localization problem in a wireless sensor network with a random topology structure. By using a specialized decomposition technique, the centralized Gauss-Newton method is modified as a distributed solution by exchanging locally information with the neighboring anchor nodes. In the recent diffusion Gauss-Newton algorithm, the main difficulty is the computation of inverse because the approximated Hessian is not always full column rank. To address this problem, we propose that two methods can be used to approximate the inverse matrix, i.e., Levenberg-Marquardt parameter and Moore-Penrose inverse methods. The former in essence is a damped iteration method, while the latter depends largely on the adjustment of step size. In addition to proposing the methods above for diffusion Gauss-Newton algorithm, we also compare their performance to solve the collaborative localization in wireless sensor networks, which can be modeled as a NLLS problem. Simulation results show both of them are applicable to diffusion GN, and LM method has some slight advantages in these respects such as resistance to environment noise while guaranteeing convergence, and dynamical increasing/decreasing to step size.