Gershgorin Analysis of Random Gramian Matrices With Application to MDS Tracking

We offer a redesigned form of the multidimensional scaling (MDS) algorithm suitable to the simultaneous tracking of a large number of targets with no a priori mobility models. First, we employ an extreme-value and asymptotic take on the theory of Gershgorin spectrum bounds to perform a detailed statistical analysis of the spectrum of random N × N Gramian matrices which arise from dynamic constructions of MDS kernels where the diagonalizer of a previous kernel is used to construct the next one. The analysis reveals that even if the subspace distance between consecutive kernels is relatively large, the dominant eigenspace of dynamic MDS kernels are, with a high probability quantified analytically, associated with its first rows. This feature is exploited further to design a statistically optimized and truncated variation of the Jacobi algorithm, which converges to the dominant eigenspace of a dynamic MDS kernel as fast as the overall optimal Jacobian, but without the exhaustive search for the elements to be annihilated at each rotation as required in the latter. Under the fact that the Euclidean double-centered kernels of the classic MDS method are asymptotically Gramian, and the knowledge of Nyström-inspired methods to compensate for data erasures, the technique presented yields a very efficient (fast) MDS-based multitarget tracking algorithm which achieves a remarkably low complexity of order O(√(N)), and which is robust to arbitrary statistics of the target´s dynamics.