An analysis of the maximum likelihood estimator for localization problems

We study the behavior of the maximum likelihood (ML) estimator for localization via triangulation, under Gaussian noise. Likelihood maximization is a non-convex problem, with possibly multiple solutions. In this paper we present an algorithm for solving this non-convex problem, which under some reasonable assumptions, is guaranteed to produce the exact ML estimate. A nice feature of our algorithm is that it is readily amenable to analysis: we give (a) a characterization of a domain in which, if the algorithm is started with an initial estimate within that domain, a standard steepest descent method is guaranteed to converge to a global minimum; (b) the distribution of the estimate; and (c) a measure of sensitivity to noise of the estimate. Many numerical examples and plots are included to illustrate these concepts.