Sparsity-Aware Extended Kalman Filter for Tracking Dynamic Graphs

TL;DR

This paper introduces a sparsity-aware extended Kalman filter for tracking evolving graph topologies from signals, effectively handling nonlinearities and noise in dynamic graph signal processing applications.

Contribution

It presents a novel EKF-based method with $ ext{l}_1$ regularization for accurately tracking sparse, time-varying graphs in a structured state-space framework.

Findings

Accurately tracks sparse, dynamic graphs under various noise conditions.

Maintains low computational complexity while handling nonlinear measurements.

Demonstrates effectiveness in realistic scenarios with changing graph structures.

Abstract

A broad range of applications involve signals with irregular structures that can be represented as a graph. As the underlying structures can change over time, the tracking dynamic graph topologies from observed signals is a fundamental challenge in graph signal processing (GSP), with applications in various domains, such as power systems, the brain-machine interface, and communication systems. In this paper, we propose a method for tracking dynamic changes in graph topologies. Our approach builds on a representation of the dynamics as a graph-based nonlinear state-space model (SSM), where the observations are graph signals generated through graph filtering, and the underlying evolving topology serves as the latent states. In our formulation, the graph Laplacian matrix is parameterized using the incidence matrix and edge weights, enabling a structured representation of the state. In order to track the evolving topology in the resulting SSM, we develop a sparsity-aware extended Kalman filter (EKF) that integrates $\ell_1$-regularized updates within the filtering process. Furthermore, a dynamic programming scheme to efficiently compute the Jacobian of the graph filter is introduced. Our numerical study demonstrates the ability of the proposed method to accurately track sparse and time-varying graphs under realistic conditions, with highly nonlinear measurements, various noise levels, and different change rates, while maintaining low computational complexity.