Algebraic Approach for Robust Localization with Heterogeneous Information

Abstract

We offer a redesigned form of the classical multidimensional scaling (C-MDS) algorithm suitable to handle the localization of multiple sources under line-of-sight (LOS) and non-line-of-sight (NLOS) conditions. To do so we propose to modify the kernel matrix used in the MDS algorithm to allow for both distance and angle information to be processed algebraically (without iteration) and simultaneously. In so doing we also show that the new formulation overcomes two well known limitations of the C-MDS approach, namely the propagation error problem and the possibility to weight the dissimilarities used as measurement information, including, for the case of binary weights, the data erasure problem. Due to the increased size of the proposed edge kernel matrix K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</sub> used in the algorithm, the Nystrom approximation is applied to reduce the overall computational complexity to few matrix multiplications. Range only scenarios are also dealt with by approximating the matrix K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</sub> . Simulations in range-angle as well as range-only scenarios demonstrate the superiority of our solution under both LOS and NLOS conditions versus semidefinite programming (SDP) formulations of the problem specifically designed to exploit the heterogeneity of the information available.