  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. However, the current Graph Signal
Processing algorithms mostly consider only the signals existing on the graph
vertices and have neglected the fact that signals can reside on the edges. 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 algorithms 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.
