Adaptive Joint Estimation of Temporal Vertex and Edge Signals
Yi Yan, Tian Xie, Erçan E. Kuruoğlu · IEEE Transactions on Signal and Information Processing over Networks · 2025
The adaptive estimation of coexisting temporal vertex (node) and edge signals on graphs is a critical task when a change in edge signals influences the temporal dynamics of the vertex signals. Existing Graph Signal Processing algorithms have extensively studied signals on graph vertices, and while recent advancements have started exploring signals on edges, a framework for systematically representing interactive time-varying signals across vertices, edges, and higher-order structures has yet to be fully realized. We propose an Adaptive Joint Vertex-Edge Estimation (AJVEE) algorithm for jointly estimating time-varying vertex and edge signals through a time-varying regression, incorporating both vertex signal filtering and edge signal filtering. Accompanying AJVEE is a newly proposed Adaptive Least Mean Square procedure based on the Hodge Laplacian (ALMS-Hodge), which is inspired by classical adaptive filters combining simplicial filtering and simplicial regression. AJVEE is able to operate jointly on the vertices and edges by merging two ALMS-Hodge specified on the vertices and edges into a unified formulation. A more generalized case extending AJVEE beyond the vertices and edges is being discussed. Experimenting on real-world traffic networks and population mobility networks, we have confirmed that our proposed AJVEE algorithm could accurately and jointly track time-varying vertex and edge signals on graphs.