On the Maximum Likelihood Approach for Source and Network Localization

We consider the source and network localization problems, seeking to strengthen the relationship between the Weighted-Least-Square (WLS) and the Maximum-Likelihood (ML) solutions of these problems. To this end, we design an optimization algorithm for source and network localization under the principle that: a) the WLS and the ML objectives should be the same; and b) the solution of the ML-WLS objective does not depend on any information besides the set of given distance measurements (observations). The proposed Range-Global Distance Continuation (R-GDC) algorithm solves the localization problems via iterative minimizations of smoothed variations of the WLS objective, each obtained by convolution with a Gaussian kernel of progressively smaller variances. Since the last (not smoothed) WLS objective derives directly from the ML formulation of the localization problem, and the R-GDC requires no initial estimate to minimize it, final result is maximum-likelihood approach to source and network localization problems. The performance of the R-GDC method is compared to that of state-of-the-art techniques such as semidefinite programming (SDP), nonlinear Newton least squares (NLS), and the Stress-of-a-MAjorizing-Complex-Objective-Function (SMACOF) algorithms, as well as to the Cramér-Rao Lower Bound (CRLB). The comparison reveals that the solutions obtained with the R-GDC algorithm is insensitive to initial estimates and provides a localization error that closely approaches that of the corresponding fundamental bounds. The R-GDC is also found to achieve a complexity order comparable to that of the SMACOF, which is known for its efficiency.