Understanding Concepts in Graph Signal Processing for Neurophysiological Signal Analysis

TL;DR

This work formalizes graph signal processing concepts for neurophysiological data, detailing two GFT definitions (filter-based and derivative-based) and their ties to total variation, PCA, and graph learning. It implements a minimalist simulated data generator to create scalable EEG-like signals with controllable spatial and spectral structure and introduces a baseline validation framework. Through perpetual cross-validation, it demonstrates that higher graph frequencies can improve classification in the simulated setting, though gains are modest and not always attributable to connectivity alone. Overall, the study highlights both the potential and current limitations of GSP for neuroimaging, underscoring the need for robust graph retrieval, validation pipelines, and careful interpretation of graph-based features.

Abstract

Multivariate signals, which are measured simultaneously over time and acquired by sensor networks, are becoming increasingly common. The emerging field of graph signal processing (GSP) promises to analyse spectral characteristics of these multivariate signals, while at the same time taking the spatial structure between the time signals into account. A central idea in GSP is the graph Fourier transform, which projects a multivariate signal onto frequency-ordered graph Fourier modes, and can therefore be regarded as a spatial analog of the temporal Fourier transform. This chapter derives and discusses key concepts in GSP, with a specific focus on how the various concepts relate to one another. The experimental section focuses on the role of graph frequency in data classification, with applications to neuroimaging. To address the limited sample size of neurophysiological datasets, we introduce a minimalist simulation framework that can generate arbitrary amounts of data. Using this artificial data, we find that lower graph frequency signals are less suitable for classifying neurophysiological data as compared to higher graph frequency signals. Finally, we introduce a baseline testing framework for GSP. Employing this framework, our results suggest that GSP applications may attenuate spectral characteristics in the signals, highlighting current limitations of GSP for neuroimaging.

Figures (7)

• Figure 1: (A) Graph with six nodes, which can be represented by an adjacency matrix $\mathbf{A}$. The displayed connections are directed, meaning that they can go in both directions. They are also weighted, which is indicated by the line width of the arrows. The graph can represent an arbitrary spatial graph topology of a multivariate signal acquired from a measurement system with six sensors. (B) Time graph with linear topology. The graph connects each time step $t_i$ to the subsequent time step $t_{i+1}$, thereby shifting a signal in time. For periodic signals, the last time step is connected to the first time step. The graph can be represented by the cyclic shift matrix $\mathbf{A}_c$
• Figure 2: Interconnections between discrete signal processing (DSP) concepts and its graph extensions. In DSP, the DFT can be traced back to the shift operator. Crucially, a derivative-based interpretation and a filter-based interpretation of the DFT are equivalent. The TV increases with increasing Fourier frequency, thereby linking the two concepts. Convolution, digital filters and finally spectral wavelets are built on the concept of the shift operator, whereby the digital filter can be expressed using the DFT. Extending the shift operator to graphs allows to build analogous concepts for graphs. Importantly, the derivative- and the filter-based GFT are not equivalent. The graph Fourier modes in both GFTs can be ordered by their frequency using either the edge-based or the node-based TV. The graph convolution can be used to define an adjacency matrix-based graph filter. The similar, Laplacian matrix-based graph filter can only be constructed by analogy and is not directly linked to the graph convolution. Graph spectral wavelets are derived from the graph convolution
• Figure 3: Lowest and highest graph Fourier modes for a geometric distance-based graph and a functional connectivity-based graph, computed from a real EEG data set. (A-C) The lowest graph Fourier modes of the geometric graph capture the fundamental symmetries and comprise the DC mode, the lateral symmetry mode and the coronal symmetry mode. (D-C) The highest graph Fourier modes are more localised. (F-H) The lowest graph Fourier modes of a functional connectivity Pearson correlation graph, which is computed from a real-world EEG data set. The graph contains negative weights, such that the DC mode is not necessarily the lowest mode. Modes 1 and 2 vaguely reflect the lateral symmetry mode 2 and the coronal symmetry mode 3 of the geometric distance-based graph. (I-J) The highest graph Fourier modes of the Pearson correlation graph are localised and still exhibit a lateral symmetry
• Figure 4: Simulated neurophysiological signals. (A) Time signals for four selected channels across the first 100 samples. Channels #2, #3 and #4 are positively, weakly and negatively correlated to channel #1, respectively. (B) Distribution of correlations of channel pairs, exhibiting the spatial connectivity structure. In the simulation, this structure is controlled by the matrix $\mathbf{A_s}$, but it is also affected by the simulation parameters. (C,D) Demonstration of power spectral density control and adjustment of classification difficulty. (C) The Fourier filter function $h$ used to colour the noise in Algorithm \ref{['alg:sim_EEG']}, assuming a sampling frequency of 256 Hz. For each data set, two conditions are simulated. Condition 2 can be varied to make it easy (solid), medium (dashed), or difficult (dotted) to distinguish from condition 1. The difficulty depends on the similarity between the two conditions. (D) Welch power spectral density of simulated signals averaged across all channels for two easily distinguishable conditions. Figures (C) and (D) clearly show that the shape of the power spectral density profile of the simulated signals can be controlled. Parameter $\alpha$ in Algorithm \ref{['alg:sim_EEG']} can be used to reduce the power density at lower frequencies
• Figure 5: Illustration of the graph frequency analysis. For each cross-validation iteration, all simulated samples from the training set are used to construct either the connectivity matrix $\mathbf{A}$ or the Laplacian matrix $\mathbf{L}$, from which the GFT matrix is computed as the eigendecomposition (see subsection \ref{['ssec:GFT']}). Using the GFT matrix, the input sample $\mathbf{X}_i$ is transformed to its GFT signal $\tilde{\mathbf{X}}_i$. The signal is further split into the 23 graph frequency signals. Lastly, a support vector machine classifier is trained on the 20 time spectral features extracted from each graph frequency signal using Welch's power spectral density method. The performance of each graph frequency can then be used to assess the quality of the spectral features in the graph frequency signal