Graph laplacian matrix learning from smooth time-vertex signal

In this paper, we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as "time-vertex signal". To realize this, we first represent the signals on a joint graph which is the Cartesian product graph of the time- and vertex-graphs. By assuming the signals follow a Gaussian prior distribution on the joint graph, a meaningful representation that promotes the smoothness property of the joint graph signal is derived. Furthermore, by decoupling the joint graph, the graph learning framework is formulated as a joint optimization problem which includes signal denoising, time- and vertex-graphs learning together. Specifically, two algorithms are proposed to solve the optimization problem, where the discrete second-order difference operator with reversed sign (DSODO) in the time domain is used as the time-graph Laplacian operator to recover the signal and infer a vertex-graph in the first algorithm, and the time-graph, as well as the vertex-graph, is estimated by the other algorithm. Experiments on both synthetic and real-world datasets demonstrate that the proposed algorithms can effectively infer meaningful time- and vertex-graphs from noisy and incomplete data.