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Graph signal processing for machine learning: A review and new perspectives
Xiaowen Dong, Dorina Thanou, Laura Toni, Michael Bronstein, Pascal Frossard
TL;DR
Large-scale structured data require effective representations and analysis methods that account for network and graph structure. This review surveys how GSP concepts and tools support machine learning through relational priors, efficiency, robustness, and interpretability, and presents future perspectives connecting signal processing with machine learning and network science.
Problem
Modern machine learning needs effective representation, processing, analysis, and visualization of large-scale structured data, especially networks and graphs.
Method
The article reviews GSP-based regularization, sampling, filters, transforms, and learning models applied to machine learning.
Results
The review finds that GSP ideas have been used to exploit relational priors and improve data efficiency, computational efficiency, robustness, and model interpretability.
Takeaways & Limitations
GSP tools provide a bridge among applied mathematics, signal processing, machine learning, and network science for complex data analysis.
Abstract
from arXiv · showhide
The effective representation, processing, analysis, and visualization of large-scale structured data, especially those related to complex domains such as networks and graphs, are one of the key questions in modern machine learning. Graph signal processing (GSP), a vibrant branch of signal processing models and algorithms that aims at handling data supported on graphs, opens new paths of research to address this challenge. In this article, we review a few important contributions made by GSP concepts and tools, such as graph filters and transforms, to the development of novel machine learning algorithms. In particular, our discussion focuses on the following three aspects: exploiting data structure and relational priors, improving data and computational efficiency, and enhancing model interpretability. Furthermore, we provide new perspectives on future development of GSP techniques that may serve as a bridge between applied mathematics and signal processing on one side, and machine learning and network science on the other. Cross-fertilization across these different disciplines may help unlock the numerous challenges of complex data analysis in the modern age.
I. INTRODUCTION
Graph signal processing (GSP) extends signal-processing concepts to graph-supported data, addressing challenges in large-scale network analysis. This review examines how GSP exploits relational structure, improves efficiency and robustness, and supports interpretability in machine learning.
- Motivation: Graph-structured machine learning supports tasks such as protein grouping and predicting gene expression over time.These tasks can help identify infection-affected genes and inform intervention strategies.
- Challenges: Large-scale biological data create demands for graph algorithms that remain robust and stable against potential noise, while interpretability matters for clinical decision-making.The introduction uses gene and protein data, including vaccine-related analysis, to illustrate these challenges.
- Data and graph representation: GSP models data as signals on weighted-graph nodes, jointly representing node attributes and underlying edge structure.It extends sampling, convolution, and frequency-domain filtering to graph signals.
- Contributions: GSP provides tools for exploiting intrinsic data structure and geometry in machine learning analysis.These tools motivate models such as graph convolutional networks and can improve classical network algorithms such as spectral clustering.
- Interpretability and broader perspective: GSP also offers perspectives on model interpretability and on inferring underlying network structure from observations.The review positions GSP as a bridge among machine learning, signal processing, and network science.
- Review scope: The review organizes GSP-based ideas around relational priors, efficiency and robustness, and interpretable machine learning systems.The discussed techniques include graph regularization, graph sampling, filters, transforms, and GSP-inspired learning models.
II. GSP FOR EXPLOITING DATA STRUCTURE
GSP generalizes signal-processing concepts to graph-supported data, using graph structure, frequency, regularization, and filtering to model relational dependencies. These tools support supervised and unsupervised learning, including multivariate regression with graph-informed response-variable interdependencies.
- Graph signal foundations: GSP models, analyzes, and processes data supported on graphs by generalizing classical signal-processing concepts to graph signals.A graph signal assigns a scalar value to each node, while more general settings can use vectors.
- Graph signal foundations: Graph Laplacian or shift-operator eigenvectors provide graph frequencies and define the graph Fourier transform.For the Laplacian decomposition, eigenvalues are conventionally sorted increasingly and serve as discrete frequencies.
- Graph signal foundations: Graph frequency supports smoothness measures, graph-based regularization, and generalized convolution and frequency-domain filtering.The graph Fourier coefficients correspond to the eigenvalue-associated discrete frequencies.
- Learning with graph structure: GSP-based tools exploit data structure and relational priors across supervised and unsupervised learning problems, with graphs either known or inferred from observations.The reviewed applications assume an underlying graph that may be predefined or learned from variables.
- Learning with graph structure: Graph regularization and graph filters can model interdependencies among multiple response variables and enrich multi-output Gaussian-process kernels.One approach learns a graph filter suited to observed graph data, improving prediction performance.
B. Graph-based classification
Graph-based classification adapts convolutional ideas to graph-supported signals through graph spectral filters, addressing the lack of a shift and convolution notion on irregular graphs. ChebNet and GCN improve locality and scalability, while GCN achieves strong semi-supervised classification results and GNNs gain GSP-based interpretability.
- Graph signal classification requires graph-specific convolution because classical CNNs lack shift and convolution notions on irregular graph domains.
- Spectral graph convolution: Spectral CNNs replace spatial filters with graph filters, multiplying graph Fourier coefficients by learnable parameters before returning to the node domain.The output is passed through a nonlinearity applied to node-wise signal values.
- Limitations and scalability: The earlier spectral CNN is limited because its filters need not be spatially localized and its eigendecomposition and graph Fourier transform are computationally expensive.The cited complexities are O(N3) for eigendecomposition and O(N2) for the matrix-vector multiplication implementing the graph Fourier transform.
- ChebNet: ChebNet uses finite polynomials of graph-Laplacian eigenvalues, guaranteeing K-hop localization while avoiding explicit eigendecomposition and graph Fourier transforms.Chebyshev approximation of a scaled Laplacian improves convolution stability.
- GCN: GCN is a K = 1 special case of ChebNet that updates node features through normalized one-hop neighborhood averaging without graph pooling.Its normalized adjacency formulation can be implemented directly in the spatial node domain.
- Results and interpretation: GCN significantly outperforms several baselines on graph-based semi-supervised learning tasks, helping spark broad interest in GNN development.GNNs can also be interpreted as learning denoising or smoothing operations on graph signals.
C. Graph-based clustering and dimensionality reduction
Graph-based clustering partitions nodes using similarity graphs, while GSP tools extend clustering to multiple scales and support related dimensionality-reduction architectures. Graph wavelets provide localized, scale-dependent node views whose correlations can drive hierarchical clustering.
- Graph-based clustering: Graph-based clustering partitions a node set into mutually exclusive clusters using a similarity graph.
- Graph-based clustering: GSP concepts support multi-scale clustering, which produces clustering results representing node groupings at different scales.
- Graph-based clustering: Graph wavelets act as scaled graph-filter outputs of node impulses, providing localized views of the graph centered at each node.
- Graph-based clustering: Correlations between graph-wavelet views yield scale-specific node similarities that can be processed with hierarchical clustering.
- Dimensionality reduction: Dimensionality reduction seeks low-dimensional representations of entities while preserving pairwise similarities encoded by a graph.
- Dimensionality reduction: GCN-based graph autoencoders and ChebNet–recurrent architectures combine GSP-inspired models with classical representation-learning methods.
III. GSP FOR IMPROVING EFFICIENCY AND ROBUSTNESS
GSP tools improve machine-learning efficiency and robustness by exploiting graph structure, especially when training data are scarce or noisy. Sampling and filtering also reduce the computational burden of graph-based learning at large scale.
- GSP-inspired models improve robustness to scarce and noisy training data or adversarial examples across several learning settings.The listed settings include supervised, semi-supervised, few-shot, zero-shot, and multi-task learning.
- GSP tools support robustness improvements in supervised, semi-supervised, few-shot, zero-shot, and multi-task learning.
- Graph sampling and filtering reduce computational complexity and time for large-scale graph-based learning.
A. Improvement on data efficiency and robustness
Graph regularization and GSP-inspired architectures improve data efficiency and robustness by encoding similarity or correlation structures among examples, representations, and tasks. These ideas support semi-supervised, few-shot, zero-shot, and multi-task learning.
- Graph regularization introduces an inductive bias toward effective learning by penalizing undesirable changes in graph-based representations.
- Smoothness regularization on similarity graphs has been shown to improve robustness in supervised and semi-supervised learning.
- GNNs improve generalization to novel few-shot tasks using only a few examples from unknown classes.
- In zero-shot learning, graph-based isoperimetric loss promotes smooth sample embeddings through polynomial approximations associated with spectral graph wavelets.
- In multi-task learning, task graphs encode correlations among tasks, and graph regularization exploits these relationships to generalize learning across tasks.
B. Robustness against topological noise
Topological noise from real-world graph uncertainty motivates stability analysis for GSP and graph-based learning. Recent results establish conditional stability for spectral filters and characterize how perturbations affect GNN outputs and network-analysis operations.
- Graph topology may be noisy or inaccurately inferred, making stability and robustness important for graph-based model generalization.
- Laplacian eigenvectors are not stable under graph-topology perturbations, motivating scrutiny of Laplacian-based spectral filters and GNN designs.
- Under certain conditions, Cayley-smoothness spectral filters are stable under absolute Laplacian perturbations, with output changes linear in perturbation magnitude.
- Spectral-domain GNN designs can transfer between graphs that discretize the same underlying metric space.
- Perturbation analyses bound GNN-output changes using graph misalignment, perturbation norms, filter smoothness, and architecture depth and width.
- Studies of random graph realizations, Laplacian spectra, and weighted-graph centrality contribute to understanding robustness against topological noise.
C. Improvement on computational efficiency
GSP tools reduce the computational burden of learning on large graphs by avoiding costly eigendecompositions, sampling fewer feature vectors, and exploiting graph factorizations for efficient processing.
- Spectral clustering scales poorly with graph size because graph construction, Laplacian eigendecomposition, and k-means each become computationally expensive.
- GSP filtering and random signal sampling simplify the last two steps of spectral clustering, reducing the number of feature vectors processed from N.
- Compressive spectral clustering bypasses eigenvector computation by filtering random graph signals and applies k-means to a small number of sampled feature vectors.
- Product-graph models factor large graphs into smaller components, enabling more efficient GFT and filtering operations.
- Cartesian product graphs can modularize regular-grid images, improving eigendecomposition efficiency through parallelization and vectorization.
IV. GSP FOR ENHANCING MODEL INTERPRETABILITY
GSP supports interpretability by inferring hidden graph structure and analyzing graph-signal properties in complex systems and learning models. These tools connect signal smoothness, filtering, and adaptive edge weights to interpretable representations and explanations.
- Topology inference estimates hidden graph structure from observed graph signals, supporting understanding of complex systems.
- GSP-based graph-learning frameworks enforce desirable signal representations through graph frequency analysis and filtering.
- Smoothness and diffusion assumptions have been applied to identify weather patterns, brain connectivity, mobility patterns, and 3D point-cloud structure.
- DNN feature regularization learns a graph over neurons and constrains intermediate features with a smoothness-promoting regularizer, increasing layer interpretability.
- GATs adaptively update edge weights, enabling interpretation of neighbor importance; GATs can also be interpreted as denoising graph edge-weight signals.
B. A posteriori interpretation of learning architectures
GSP provides a posteriori interpretations of learning architectures by treating task-specific hypotheses and intermediate representations as graph signals. The reviewed approaches use signal fluctuations or smoothness to explain predictions and assess class separation, while noting scalability limits.
- MARGIN constructs a graph of task entities and represents a task hypothesis as a signal carrying semantic information.
- MARGIN uses GSP to identify graph-signal fluctuations, with high-pass filtering highlighting nodes that describe label variations.
- MARGIN computes node influence from the magnitude of filtered signal values and translates those scores into interpretable explanations.
- MARGIN becomes computationally intensive when the entire dataset is selected as the domain because graph construction and filtering scale with dataset size.
- A DNN-monitoring framework uses label-signal smoothness on layer-wise k-nearest-neighbor graphs as a measure of class separation in intermediate representations.
- Future interpretability analyses could examine local smoothness, sparsity, centrality, and modularity alongside global smoothness.
A. GSP and probabilistic modeling
The paper connects GSP with probabilistic graph modeling and decision-making under uncertainty. It describes Bayesian graph-learning directions and graph-signal representations for reinforcement learning, while identifying topology uncertainty, computational cost, and high-dimensional online control as open challenges.
- A promising direction combines GSP, machine learning, and probabilistic graph-topology modeling in a coherent framework.
- Real-world graph topologies are often noisy, incomplete, or evolving, motivating analysis that explicitly accounts for topology uncertainty.
- An Erdös-Rényi model has been used to represent topological noise and analyze its effect on graph-signal filtering and ICA.
- The reviewed Bayesian GCN work models graphs with a mixed-membership stochastic block model and computes parameter posteriors for graph generation.
- In decision-making problems, graph states or contexts and rewards can be represented as graph signals, allowing sparse representations and lower-dimensional learning.
- GSP-based sparse representations can improve learning rates in large state spaces, where unscaled learning otherwise yields highly suboptimal decisions.
- Decision-making systems commonly require costly Laplacian inversion or optimization at each decision opportunity, motivating lower-complexity GSP tools.
- Existing high-dimensional network optimization techniques do not directly apply to adaptive online sequential decisions, leaving online control under uncertainty open.
C. GSP and model interpretability
GSP links graph structure and relational priors to more explainable learning, addressing the non-robust, black-box behavior of deep models in safety-critical settings. Extensions to higher-order structures may further improve interpretability beyond pairwise interactions.
- Motivation: Deep learning on large systems typically produces non-robust, black-box decisions, creating difficulties for safety-critical applications such as personalized medicine and autonomous systems.The paper identifies interpretability as a central concern in these settings.
- Graph structure and relational learning: Graph modeling incorporates domain knowledge such as physical interactions and biases learning toward relational representations.This relational structure is presented as a basis for more human-like intelligence.
- Graph structure and relational learning: GNN architectures that incorporate graph structure are described as more explainable than typical black-box models.The paper connects this explainability to architectures designed around relational structure.
- Future directions: Extending GNNs toward more generic permutation-invariant functions is presented as a route to further advances in relational learning and interpretability.The passage frames this as a prospective extension rather than an established result.
- Higher-order structures: Higher-order structures, including motifs, simplicial complexes, and hypergraphs, extend analysis beyond pairwise interactions and may support more interpretable machine-learning models.Topological signal processing is identified as a promising first step, while motifs have also been used in directed-graph GNNs and community detection.