Methods of Adaptive Signal Processing on Graphs Using Vertex-Time Autoregressive Models

TL;DR

This paper introduces an adaptive filtering approach for identifying the topology of random graph processes using vertex-time autoregressive models, enabling online estimation and analysis of graph signals.

Contribution

It develops an online adaptive filtering method based on stochastic gradient projection for topology identification in random graph processes, extending previous batch methods.

Findings

Effective topology recovery demonstrated in experiments

Algorithm converges under certain conditions

Outperforms traditional methods in dynamic scenarios

Abstract

The concept of a random process has been recently extended to graph signals, whereby random graph processes are a class of multivariate stochastic processes whose coefficients are matrices with a \textit{graph-topological} structure. The system identification problem of a random graph process therefore revolves around determining its underlying topology, or mathematically, the graph shift operators (GSOs) i.e. an adjacency matrix or a Laplacian matrix. In the same work that introduced random graph processes, a \textit{batch} optimization method to solve for the GSO was also proposed for the random graph process based on a \textit{causal} vertex-time autoregressive model. To this end, the online version of this optimization problem was proposed via the framework of adaptive filtering. The modified stochastic gradient projection method was employed on the regularized least squares objective to create the filter. The recursion is divided into 3 regularized sub-problems to address issues like multi-convexity, sparsity, commutativity and bias. A discussion on convergence analysis is also included. Finally, experiments are conducted to illustrate the performance of the proposed algorithm, from traditional MSE measure to successful recovery rate regardless correct values, all of which to shed light on the potential, the limit and the possible research attempt of this work.