Graph Error Effect in Graph Signal Processing

Abstract

The first step for any graph signal processing (GSP) procedure is to learn the graph signal representation, i.e., to capture the dependence structure of the data into an adjacency matrix. Indeed, the adjacency matrix is typically not known a priori and has to be learned. However, it is learned with errors. A little, if any, attention has been paid to modeling such errors in the adjacency matrix, and studying their effects on GSP methods. However, modeling errors in adjacency matrix will enable both to study the graph error effects in GSP and to develop robust GSP algorithms. In this paper, we therefore introduce practically justifiable graph error models. We also study, both analytically and in terms of simulations, the graph error effect on the performance of GSP methods based on the examples of more traditional different types of filtering of graph signals and less known independent component analysis (ICA) of graph signals (graph decorrelation).

Robust Graph-Filter Identification with Graph Denoising Regularization

“…Modeling graph perturbations. A worth discussing topic is the postulation and analysis of graph-perturbation models that combine practical relevance with analytical tractability [22]. Unfortunately, space limitations prevent us from engaging in that discussion and we limit ourselves to describe one model, which will motivate the formulation in the next section.…”

“…Despite their theoretical and practical relevance, the number of works dealing with robust GSP approaches is limited [21][22][23]. Using a small perturbation analysis, [21] first studies how link imperfections affect the spectrum of the graph Laplacian and then proposes a Bayesian framework.…”

“…Using a small perturbation analysis, [21] first studies how link imperfections affect the spectrum of the graph Laplacian and then proposes a Bayesian framework. The focus of [22] is on postulating (graphon-based) perturbation models and analyzing how those perturbations affect graph-signal operators. Differently, [23] combines structural equation models (SEMs), which can be viewed as a particularization of the GSP framework, with total least squares (TLS) for graph signal inference while also inferring the perturbations of the observed graph.…”

Robust Least Mean Squares Estimation of Graph Signals

“…This is repeated until the adjacency matrix estimate does not change. The sampling is done using one of the graph error models presented in [4]. The model is written as…”

“…In this paper, we consider recovering a graph signal, when the presumed graph differs from the underlying graph. The effect of such graph error has been of increasing interest recently (see [4,5,6]), and as far as we know, the first robust methods with respect to errors in graph topology was proposed in [7], where graph signal spectral analysis and clustering were considered. However, the authors of [7] assume knowledge of edgewise probabilities of perturbation, which differs from our approach as we only rely on the estimate of the graph.…”