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Adaptive Joint Estimation of Temporal Vertex and Edge Signals

Yi Yan, Tian Xie, Ercan E. Kuruoglu

TL;DR

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.

Abstract

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 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.

Adaptive Joint Estimation of Temporal Vertex and Edge Signals

TL;DR

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.

Abstract

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 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.
Paper Structure (17 sections, 50 equations, 9 figures, 2 tables, 2 algorithms)

This paper contains 17 sections, 50 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary knowledge
  3. Graph representation
  4. Signal Processing on graphs and simplicial complexes
  5. The ALMS-Hodge
  6. Adaptive joint vertex-edge estimation
  7. Vertex regression influenced by time-varying edge signals
  8. Joint estimation of vertex and edge signals
  9. Generalizing AJVEE to high-ordered simplicial complexes
  10. Experiment Results and Discussion
  11. Experiment settings
  12. ALMS-Hodge analysis
  13. Joint Estimation on transportation network
  14. COVID-19 case forecasting in population mobility network
  15. Conclusion, Limitation, and Future Work
  16. ...and 2 more sections

Figures (9)

  • Figure 1: The Sioux Falls network with time-varying signals on both the vertices and the edges. (left color bar: edges, right color bar: vertices)
  • Figure 2: Performance of the ALMS-Hodge under fixed noise and different step sizes.
  • Figure 3: Performance of the ALMS-Hodge under fixed step size and different noise levels.
  • Figure 4: Estimation on one of the unobserved edges (top) and one of the unobserved vertices (bottom) of the Sioux Falls network.
  • Figure 5: NMSE of the AJVEE compared against separate estimation in the Sioux Falls network. Top: edges. Bottom: vertices.
  • ...and 4 more figures