Joint Graph Learning and Signal Recovery via Kalman Filter for Multivariate Auto-Regressive Processes

Abstract

In this paper, an adaptive Kalman filter algorithm is proposed for simultaneous graph topology learning and graph signal recovery from noisy time series. Each time series corresponds to one node of the graph and underlying graph edges express the causality among nodes. We assume that graph signals are generated via a multivariate auto-regressive processes (MAR), generated by an innovation noise and graph weight matrices. Then we relate the state transition matrix of Kalman filter to the graph weight matrices since both of them can play the role of signal propagation and transition. Our proposed Kalman filter for MAR processes, called KF-MAR, runs three main steps; prediction, update, and learn. In prediction and update steps, we fix the previously learned graph weight matrices and follow a regular Kalman algorithm for graph signal recovery. Then in the learning step, we use the last update of graph signal estimates and keep track of topology changes. Simulation results show that our proposed graph Kalman filter outperforms the available online algorithms for graph topology inference and also it can achieve the same performance of the batch method, when the number of observations increase.