Abstract
In sensor networks, adaptive algorithms such as diffusion adaptation LMS and RLS are commonly used to learn and track non-stationary signals. When such signals have similarities across certain nodes as captured by a graph, then Laplacian Regularized (LR) LMS and diffusion adaptation LR LMS can be utilized for the respective centralized and distributed estimation cases. What if the ground truth signal’s time-varying co-variance structure is related to a time-varying graph? And what if there exists outlier/anomaly nodes trying to influence the graph signal? In order to answer these questions, we first re-examine the existing adaptive methods, and use graph signal processing notions to augment the algorithms with an additional graph filtering step for regularization. We also study a distributed adaptive algorithm based on message passing that does not require any global information and scales to large time-varying graphs. In particular, each node augments adaptive filtering steps with an additional local filtering steps based on a Local Graph Transform (LGT) defined by the particular node’s local graph Laplaican. Moreover, we demonstrate how to design these graph filters, leading to performance improvements over existing methods. We also analyze the stability and convergence of our methods and illustrate how the empirical performance is captured by the theoretical results which unveil the bias and variance trade-off. Finally, we examine the problem of estimating and tracking non-stationary signals and outliers in noisy streaming data emanating from both static and time-varying graphs. In conjunction with adaptive algorithms and optimization methods, we in- corporate the LGT approach and outlier estimation. Through simulations and theoretical performance analysis we demonstrate the efficacy of this LGT-based approach, which is scalable and suitable for handling large time-varying graphs.