Local strong convexity of maximum-likelihood TDOA-Based source localization and its algorithmic implications

We consider the problem of single source localization using time-difference-of-arrival (TDOA) measurements. By analyzing the maximum-likelihood (ML) formulation of the problem, we show that under certain mild assumptions on the measurement noise, the estimation errors of both the closed-form least-squares estimate proposed in [1] and the ML estimate, as measured by their distances to the true source location, are of the same order. We then use this to establish the curious result that the objective function of the ML estimation problem is actually locally strongly convex at an optimal solution. This implies that some lightweight solution methods, such as the gradient descent (GD) and Levenberg-Marquardt (LM) methods, will converge to an optimal solution to the ML estimation problem when properly initialized, and the convergence rates can be determined by standard arguments. To the best of our knowledge, these results are new and contribute to the growing literature on the effectiveness of lightweight solution methods for structured non-convex optimization problems. Lastly, we demonstrate via simulations that the GD and LM methods can indeed produce more accurate estimates of the source location than some existing methods, including the widely used semidefinite relaxation-based methods.