Graph laplacian matrix learning from smooth time-vertex signal

• Published in: China communications Vol. 18; no. 3; pp. 187 - 204
• Main Authors: Li, Ran, Wang, Junyi, Xu, Wenjun, Lin, Jiming, Qiu, Hongbing
• Format: Journal Article
• Language: English
• Published: China Institute of Communications 01.03.2021; School of Telecommunications Engineering, Xidian University, Xi'an 710071, Shanxi, China; Key Laboratory of Cognitive Radio and Information Processing, Ministry of Education (Guilin University of Electronic Technology), Guilin 541004, Guangxi, China%Key Laboratory of Cognitive Radio and Information Processing, Ministry of Education (Guilin University of Electronic Technology), Guilin 541004, Guangxi, China%School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China
• Subjects: Cartesian product graph; Correlation; Covariance matrices; discrete second-order difference operator; Gaussian prior distribution; graph Laplacian matrix learning; Laplace equations; Signal processing algorithms; Sparse matrices; Spatiotemporal phenomena; spatiotemporal smoothness; Time-domain analysis; time-vertex signal; spatiotemporal smoothness; graph Laplacian matrix learning; Cartesian product graph; discrete second-order difference operator; Gaussian prior distribu-tion; time-vertex signal
• ISSN: 1673-5447
• DOI: 10.23919/JCC.2021.03.015

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.