Graph Signal Processing Framework

• Graph Signal Processing is a framework for analyzing signals on irregular graph structures using spectral decompositions and advanced filtering techniques.
• It employs operators like the Laplacian and tensor-based shift operators to enable graph Fourier transforms, sampling, and reconstruction, impacting diverse domains such as neuroscience and energy grids.
• Recent advances extend GSP to simplicial complexes, multilayer graphs, and probabilistic models, enhancing scalability, robustness, and performance in high-dimensional and uncertain networks.

Graph Signal Processing (GSP) refers to a unified theoretical and algorithmic framework for analyzing, transforming, and processing structured signals defined on the vertices of a graph or, more generally, on combinatorial objects encoding network relationships. Unlike classical signal processing, which operates on regularly ordered domains (lines, grids, tori), GSP exploits the discrete, irregular, and sometimes multi-dimensional topology of networked data to formulate analogues of Fourier analysis, filtering, sampling, convolution, and prediction. Modern GSP subsumes classical DSP-on-graphs (Sandryhaila et al., 2012), incorporates harmonic analysis on simplicial complexes (Ji et al., 2020), accommodates multilayer and tensorized domains (III et al., 2020, Zhang et al., 2021), allows for community and distributional perspectives (Petrovic et al., 2020, Ji et al., 2020, Ji et al., 2021, Zhao et al., 30 Sep 2025, Ji et al., 2023), and has led to robust and scalable toolboxes (Perraudin et al., 2014) impacting domains from neuroscience (Goerttler et al., 2023) to energy grids (Ramakrishna et al., 2021) and social networks (Zhang et al., 2019).

1. Foundational Structures: Signals, Operators, and Domains

Central to the GSP formalism is the assignment of a signal to a combinatorial structure—typically, a weighted graph $G=(V,E,W)$ with $|V|=N$ nodes and weight matrix $W\in\mathbb{R}^{N\times N}$, or a generalization such as a simplicial complex $X$ or a multilayer/hypergraph tensor (Ji et al., 2020, Zhang et al., 2021, Zhang et al., 2019).

A graph signal is a vector $x \in \mathbb{R}^N$, assigning a value to each node. For higher-order domains, the signal may be defined on $k$-simplices (edges, triangles, tetrahedra) or as tensor-valued objects $X\in\mathbb{R}^{N_1\times N_2\cdots N_D}$ (III et al., 2020).

The choice of shift operator (adjacency $A$, Laplacian $L$, modularity $B$, boundary $\partial_k$, or generalized tensors) encodes the structural regularities and local relationships crucial for defining frequency, smoothness, and modes of propagation (Sandryhaila et al., 2012, Ji et al., 2020, Petrovic et al., 2020).

The spectral decomposition of these operators, most commonly via the eigendecomposition $L=U\Lambda U^T$ or tensor CP decomposition (HGSP), establishes the graph Fourier modes and spectral basis.

2. Graph Fourier Analysis and Spectral Filtering

The analog of the Fourier transform in GSP is the graph Fourier transform (GFT) (Sandryhaila et al., 2012, Goerttler et al., 2023, Perraudin et al., 2014). For Laplacian-based frameworks, $x$ is decomposed into graph frequencies by projecting onto the eigenbasis:

$\hat{x} = U^T x$

with $U$ orthonormal eigenvectors, $\Lambda = \text{diag}(\lambda_1,\ldots,\lambda_N)$ graph frequencies. Low $\lambda$ signify globally smooth modes; high $\lambda$ encode fine, high-variation structures.

Spectral filtering is performed by applying spectral multipliers or window functions $g(\lambda)$, redefining classical filtering operations (low-pass, band-pass, wavelets) on graphs. The filtered signal is

$y = U\,g(\Lambda)\,U^T x$

avoiding explicit diagonalization via polynomial approximations, like Chebyshev expansions, for scalability (Perraudin et al., 2014, Goerttler et al., 2023).

Extensions include spectral analysis for:

• Simplicial complexes: $L_k$-based GFT for $k$-simplices (Ji et al., 2020)
• Multilayer graphs: tensor eigen-decomposition and joint M-GFT (Zhang et al., 2021)
• Hypergraphs: orthogonal-CP (E-eigenpair) HGFT (Zhang et al., 2019)
• Distribution-valued signals: Wasserstein pushforward GFT (Zhao et al., 30 Sep 2025)
• Community-aware modes: modularity-matrix GFT (Petrovic et al., 2020)

3. Generalizations: High-Dimensional, Tensor, and Probabilistic GSP

Classical GSP is restricted to signals on single-layer (pairwise) graphs. Recent frameworks leverage:

• Simplicial complexes: Unify vertex, edge, face signals and their Laplacians, recovering classical GSP at $k=0$, and providing edge/face circulation modes, smoother anomaly detection, and more accurate label denoising in citation networks (gains of 5–10% over Laplacian filtering) (Ji et al., 2020).
• Multilayer graphs: Tensor-based M-GSP encodes multi-level interactions (IoT, RGB images). Joint M-GFT diagonalizes filtering on supra-adjacency (Zhang et al., 2021).
• Multi-way tensors: MWGSP generalizes MGFT to tensors by mode-wise Kronecker combinations. This enables efficient denoising, energy compaction, and spatiotemporal filtering in high-dimensional data, outperforming mode-wise or matrix-only approaches (III et al., 2020).
• Hypergraph signal processing (HGSP): Models $n$-ary interactions, introduces HGFT, and achieves superior compression and clustering metrics (e.g., 1.47× average ratio versus GSP) (Zhang et al., 2019).
• Probabilistic/distributional GSP: Models uncertainty in topology or signals, replacing fixed operators by probability distributions, and defines expectation-based Fourier and filtering (distributional or mean-filtered convolution, MFC) (Ji et al., 2020, Ji et al., 2021, Ji et al., 2023, Zhao et al., 30 Sep 2025).
• Hilbert-space GSP: Encodes infinite-dimensional (e.g., continuous-time or multichannel) signals per node. Joint stationarity, spectral forms, and Wiener estimation generalize classical GSP to arbitrary separable Hilbert spaces (Ji et al., 2019, Jian et al., 2021).

4. Sampling, Reconstruction, and Locality Principles

Sampling and reconstruction on graphs generalize the Shannon paradigm, with bandlimited signals defined in spectral (GFT) or probabilistic terms. Key algorithms include:

• Bandlimited sampling: Recovery of $K$-bandlimited signals from $K$ point samples when the subsampled eigenbasis is full-rank—direct analog of Nyquist with optimal placement via maximizing singular values (Perraudin et al., 2014, Ramakrishna et al., 2021).
• Multilayer/tensor sampling: Block-Kronecker and tensor methods enable mode-by-mode sampling, with computational gains O($D N_m^3$) (III et al., 2020, Zhang et al., 2021).
• Distributional (ensemble) recovery: Sampling and recovery in expectation with convexity or $\epsilon$-bandlimitedness guarantees, outperforming any single operator choice during network uncertainty (Ji et al., 2020).
• Local distribution framework: Polynomial graph filters depend only on $K$-hop neighborhoods. Filter transferability and spectral density convergence established via Wasserstein distance on local rooted-ball measures (Roddenberry et al., 2022).

5. Extensions, Applications, and Computational Aspects

The GSP paradigm has been extended and deployed in myriad domains:

• GSPBox: Provides optimized implementations (MATLAB, Python) of graph construction, Laplacian types, spectral transforms, wavelet/bandpass filters, and optimization wrappers for convex problem-solving (Perraudin et al., 2014).
• Grid-GSP: Models power grids as graphs with admittance Laplacians; voltage data are low-pass graph-filtered, enabling scientifically interpretable bandlimited sampling, anomaly detection (false data injection), network inference, and graph-based compression (Ramakrishna et al., 2021).
• Neurophysiology: GSP decomposes EEG/MEG/fMRI signals in spatial graph frequencies; classification accuracy in high-frequency bands is marginally superior but true anatomical connectivity is only weakly exploited, motivating need for improved graph-aware models (Goerttler et al., 2023).
• Community-aware GSP: Modularity-matrix filtering, sampling, surrogates, and denoising lead to improved within-community coherence, error reduction, and uncover unique behavioral links in neuroimaging (Petrovic et al., 2020).
• Signal classification/compression: Blog (customer) labeling, weather-station temperature compression, and linear prediction are unified under discrete DSP-on-graphs FIR filtering (Sandryhaila et al., 2012).
• Distributed and categorical GSP: Abstract message passing algorithms enable privacy-preserving, non-iterative convex optimization across distributed subgraphs, generalizing message forms and local solubility (Ji et al., 2022). Category-theoretic GSP identifies correspondences underpinning uncertainty and compositional filtering (Ji et al., 2023).
• Graphon Signal Processing (GnSP): Spectral analysis and filtering over continuum graphons provide stable, scalable, trial-invariant embeddings for spiking and biological neural networks, with spectral convergence and robustness rigorously established (Sumi et al., 24 Aug 2025).

6. Theoretical Impact and Future Directions

The graph signal processing framework profoundly generalizes signal processing to irregular, multiscale, and uncertain data domains. Key theoretical advances include:

• Unified treatment of vertex, edge, and higher-order signals via Hodge Laplacians and tensor spectrum (Ji et al., 2020, Zhang et al., 2019).
• Probabilistic operator spaces, categorical uncertainty, and Wasserstein-distribution signals allow robust, flexible modeling in real-world settings (Ji et al., 2020, Ji et al., 2021, Zhao et al., 30 Sep 2025, Ji et al., 2023).
• Joint stationarity in generalized Hilbert spaces—new power spectral density forms for estimation and completion (Jian et al., 2021, Ji et al., 2019).
• Locality and transferability principles via empirical rooted-ball distributions (Roddenberry et al., 2022).
• Unification and extension to deep learning architectures for local, non-local, and adaptive convolutional filtering (Puy et al., 2017, III et al., 2020).

Ongoing challenges address computational scalability for high-dimensional and streaming domains, principled graph and filter learning, multivariate spectral kernel design, extensions to directed/hyper/graphon objects, and interpretability across scientific applications.

7. Comparative Table: Generalizations in Graph Signal Processing

Generalization	Core Operator	Signal Domain	Spectral Theory
Classical GSP	Laplacian, Adjacency	$\mathbb{R}^N$	Eigen-decomposition
Simplicial Complexes (Ji et al., 2020)	Hodge Laplacians	$\mathbb{R}^{|X_k|}$ for $k$-simplices	Orthonormal basis of $L_k$
Multilayer Graphs (Zhang et al., 2021)	Adjacency/Laplacian Tensors	$\mathbb{R}^{M \times N}$	Tensor Eigen-decomposition
Multi-way Tensors (III et al., 2020)	Product Laplacians	$\mathbb{R}^{N_1\times...\times N_D}$	Kronecker Eigenbasis
Hypergraph SP (Zhang et al., 2019)	Laplacian Tensors	Outer products, tensors	Orthogonal-CP decomposition
Probabilistic/Distributional (Ji et al., 2020, Zhao et al., 30 Sep 2025)	Operator Distribution	Probability on $\mathbb{R}^N$	Ensemble spectral averaging
Generalized/Hilbert Space (Ji et al., 2019, Jian et al., 2021)	Tensor Product Operators	$\mathbb{C}^n \otimes H$	Joint spectral measure
Graphon SP (Sumi et al., 24 Aug 2025)	Integral Operator	$L^2([0,1])$	Compact operator spectrum

Classical frameworks form special cases within each generalization, recovering standard DSP-on-graphs formulas at appropriate limits or when higher-order/nondeterministic structure is absent.

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Graph signal processing stands as a mathematically coherent, extensible, and empirically validated framework for analyzing signals on complex networked data structures. The formal unification of spectral, sampling, and filtering tools with combinatorial and probabilistic structures continues to drive progress in data science, applied mathematics, and network modeling.