Graph Signal Processing: Foundations & Applications

• Graph Signal Processing (GSP) is a framework that generalizes traditional digital signal processing to data on graphs, enabling analysis and filtering on irregular network structures.
• GSP leverages mathematical foundations like graph Fourier transforms, polynomial filters, and spectral decompositions to quantify signal variation with respect to graph topology.
• GSP underpins diverse applications in neuroimaging, infrastructure resilience, machine learning, and geometric data analysis, offering scalable tools for signal reconstruction and anomaly detection.

Graph Signal Processing (GSP) is a framework that extends classical digital signal processing (DSP) concepts to datasets defined on graphs, enabling the analysis, filtering, and transformation of signals indexed by nodes in complex, irregular networks. GSP generalizes core DSP operations—such as shift, convolution, and Fourier transformation—by leveraging graph topology encoded by matrices such as the adjacency or Laplacian operator. This allows the exploitation of the inherent relational structure of data encountered in a wide range of domains, including sensor networks, biological systems (e.g., neural imaging), machine learning, infrastructure networks, image and 3D geometric data, and beyond.

1. Mathematical Foundations and Core Principles

At the heart of GSP is the generalization of "shift" and "frequency" to the graph domain. In the time-series case, the shift operator is the circulant delay matrix, and frequency arises from its eigenvectors (the classical Fourier basis). In GSP, shift operators are typically chosen as the adjacency matrix $A$ or the (combinatorial or normalized) Laplacian $L = D - A$, where $D$ is the degree matrix. These operators are diagonalizable:

$S = U \Lambda U^{-1}$

where $S$ can be $A$ or $L$, $U$ is the eigenvector matrix, and $\Lambda$ is the diagonal matrix of eigenvalues, termed "graph frequencies." The Graph Fourier Transform (GFT) is then defined as:

$\hat{x} = U^{-1} x \quad\text{(adjacency-based)}$

$$\hat{x} = U^T x \quad\text{(symmetric/Laplacian-based)}$$

and enables spectral representation and analysis of the graph signal $x \in \mathbb{R}^N$.

The quadratic form $$x^T L x = \sum_{(i,j)\in \mathcal{E}} w_{ij} (x_i - x_j)^2$$ quantifies the signal's variation with respect to the graph structure, and low eigenvalues (low graph frequencies) correspond to smooth functions on the graph.

Graph filters act as polynomials in the shift operator (e.g., $h(L) = \sum_{k=0}^{K-1} h_k L^k$), promoting spatial or spectral localization depending on their construction. Filtering and convolution can equivalently be performed in the frequency domain:

$$\text{If } x \star g = U \operatorname{diag}(\tilde{g}) U^{-1} x$$

where $\tilde{g}$ is the filter's spectral response.

2. Extensions: Generalized and Vector-Valued Frameworks

Classical GSP treats scalar signals at each node; several extensions generalize this paradigm:

• Vector-Valued Signals: Signals are modeled as functions $x: V(G) \to \mathcal{X}$, with $\mathcal{X}$ a Banach or Hilbert space (e.g., $\mathbb{R}^d$, $L^2(\Omega)$). The signal space $X^N$ admits an expanded vertex-frequency analysis, with generalized Fourier transforms, convolution, and translation operators defined by hierarchical or tensor-product bases. These provide uncertainty principles and operator norm estimates that depend on graph size and coherence of the chosen eigenbasis (Caputo, 28 May 2025, Ji et al., 2019).
• Hilbert Space Theory: Embedding GSP in separable Hilbert spaces permits rigorous definition of joint transforms and filters, crucial for multichannel, continuous, or time-varying signals at each vertex (Ji et al., 2019, Jian et al., 2021).
• Wide-Sense Stationarity (WSS): The concept of stationarity is extended to the joint graph-Hilbert space, leading to spectral decompositions where the covariance operator is diagonalized in the joint Fourier basis. This enables explicit Wiener filter formulas for denoising and reconstruction in the joint vertex-frequency domain (Jian et al., 2021).

3. GSP on Complex Data Structures: Tensors and Manifolds

Increasingly, data are represented as tensors (multi-way arrays) or samples from (possibly nonlinear) manifolds:

• Multi-way Graph Signal Processing: For tensor-valued data $X \in \mathbb{R}^{n_1 \times \cdots \times n_D}$ with a graph in each mode, product graphs (e.g., Cartesian) are constructed to encode global structure. Multi-Way Graph Fourier Transforms (MWGFT) utilize Kronecker products of modal bases, yielding scalable computation and joint regularization strategies (III et al., 2020).
• Geometric Data: GSP underpins the analysis of point clouds, images, and dynamic scenes by modeling data points as graph vertices and constructing Laplacians from Euclidean or geodesic distances. The GSP framework captures both intrinsic and extrinsic geometry, supporting filtering, restoration, and compression on highly irregular domains (Hu et al., 2020).

4. Applications in Neuroscience, Infrastructure, and Machine Learning

GSP has been widely applied to neuroimaging, engineered networks, and machine learning:

• Neuroimaging: GSP exploits the graph structure of brain areas (from geometric or functional connectivity) and represents BOLD or EEG signals as signals on brain graphs. GFT decompositions reveal aligned (low-frequency) and liberal (high-frequency) components relative to anatomical connectivity, with high-frequency modes found to be more discriminative for classification than low-frequency ones. Mixed graphs combining geometry and function, particularly semilocal constructions, have yielded superior classification and dimensionality reduction accuracy compared to classical PCA and ICA (Ménoret et al., 2017, Huang et al., 2017, Goerttler et al., 2023).
• Infrastructure Resilience: Power grids, water networks, and other infrastructures are modeled as graphs; GSP enables detection of anomalies, network inference, and sensor placement by exploiting the compressibility and spectral structure of signals such as voltages or flow measurements. Data-driven tuning of edge weights may improve signal compressibility and filtering performance (Schultz et al., 2020, Ramakrishna et al., 2021).
• Machine Learning: GSP tools—filters, transforms, wavelets—provide powerful means to exploit relational priors, improve computational/data efficiency, and enhance model interpretability. They are foundational for graph convolutional networks, spectral clustering, graph-based semi-supervised learning, and have been shown to support probabilistic graph modeling, decision-making, and uncertainty-aware architectures (Dong et al., 2020, Ortega et al., 2017).

5. Methodological Innovations: Modulation, Sampling, and Community Structure

Recent research in GSP has revealed several methodological advancements:

• Modulation and Dual Shift Operators: A duality is established between vertex- and frequency-domain operations, introducing spectral shift operators and "spectral graphs" for defining convolution, modulation, and sampling on arbitrary or directed graphs. Properly designed modulation allows frequency division multiplexing of graph signals (Shi et al., 2019, Shi et al., 2023).
• Sampling Theory: Graph sampling theorems guarantee perfect recovery of bandlimited graph signals under broad conditions. Sampling strategies may operate in both the vertex and frequency domains, enabling recovery via inversion of linear systems structured by the GFT or spectral shift. Sampling sets need not be uniform and depend critically on the GFT structure (Ortega et al., 2017, Shi et al., 2019, Giraldo et al., 2022).
• Community-Aware GSP: By using the modularity matrix in place of the Laplacian as the shift operator, community-aware GSP enables operations (filtering, sampling, denoising) that target mesoscale network structure, providing enhanced performance over Laplacian-based methods in systems with pronounced clustering (air traffic, neuroimaging) (Petrovic et al., 2020).

6. Graphon Signal Processing and Scalability

Traditional GSP methods face limitations in scalability and sensitivity to stochastic network variability. Graphon Signal Processing (GnSP) generalizes GSP to the limit of large-scale networks by considering the continuum limit (graphon) of a sequence of graphs. The spectrum of the graphon kernel operator provides stable, trial-invariant features for spectral embeddings of signals in large or random networks, with demonstrated robustness for classification and functional inverse problems in both simulated and biological neural data. This approach mitigates issues of spectral instability and enhances cross-network comparability (Sumi et al., 24 Aug 2025).

7. Open Problems and Future Directions

GSP continues to evolve in several dimensions:

• Time-varying and dynamic graphs: Handling signal and graph co-evolution remains a challenge, particularly for dynamic brain connectomics and evolving social networks (Jian et al., 2021).
• Higher-order and non-pairwise interactions: Extending GSP to data supported on hypergraphs, simplicial complexes, and multi-relational structures is an active research frontier (Dong et al., 2020).
• Integration with learning: GSP principles are being combined with deep learning, probabilistic inference, and uncertainty quantification for advanced signal representation, inference, and model design (Dong et al., 2020, Ortega et al., 2017).
• Computational scalability: Graphon-based frameworks, efficient Kronecker and multilinear algorithms, and companion models supporting fast FFT-based convolution are critical for large-scale and high-dimensional data analysis (Sumi et al., 24 Aug 2025, III et al., 2020, Shi et al., 2023).
• Interpretability and model selection: Geometry-aware analysis, modularity-informed methods, and smoothness orderings beyond classic frequency provide avenues for physically and contextually meaningful representations (Ji et al., 2022, Petrovic et al., 2020).

GSP's impact is evident across domains from neuroimaging and infrastructure to machine learning and geometry processing, driven by its ability to rigorously combine algebraic, spectral, and topological insights for data defined on complex, irregular structures.