Non-linear embedding of acoustic cross-spectral density matrices through diffusion maps

Based on the notion of similarity or “distance” between cross-spectral density matrices (CSDMs), a recent analysis of matched-field source localization in a stochastic ocean waveguide provided evidence that geodesic distances between CSDMs could be employed to estimate the source location in range and depth. For M acoustic sensors configured as a vertical array, these $M×M$ matrices were estimated from source and replica fields propagated to the array and interpreted as points in a Riemannian manifold whose dimension is M². Because they serve as fundamental constructs for many source localization algorithms, visualizations of CSDM manifolds are illustrated here in an attempt to gain insight into this geometric approach by using simulated acoustic fields propagated through an ocean waveguide with internal wave-induced variability. The manifold is treated as an undirected, weighted graph whose nodes are CSDMs with edges (weights) describing a measure of similarity between nodes. A non-linear dimensionality reduction technique, diffusion maps, is applied to project these high-dimensional matrices onto a three-dimensional subspace using a spectral decomposition of the graph in an attempt to grasp relationships among such matrices. The mapping is designed to preserve the notion of distance between matrices, allowing for a meaningful visualization of the high-dimensional manifold.