Multiview Graph Learning

Updated 16 December 2025

Multiview graph learning is a framework for jointly inferring multiple graph structures from multi-modal data, capturing both shared and unique relationships.
The approach utilizes consensus and complementary regularization schemes to enhance clustering, classification, and feature selection in high-dimensional applications.
Advanced computational methods such as block coordinate minimization, ADMM, and anchor-based strategies ensure scalability and offer theoretical guarantees.

Multiview graph learning is a research area focused on simultaneously learning multiple graph structures (and—optionally—joint node or graph representations) from data presenting several "views" or modalities, with the goal of leveraging both consensus and complementary information across these views. The field addresses challenges ranging from clustering and feature selection in multi-modal datasets to robust graph inference in high-dimensional settings such as neuroscience. This article outlines the principal formulations, algorithms, and theoretical advances in multiview graph learning, with references to key contributions in recent literature.

1. Mathematical Formulations and Problem Types

Multiview graph learning is generally posed as a joint optimization over $V$ graphs $G^{(v)}$ or their Laplacians $L^{(v)}$ for a common set of $n$ entities, possibly along with additional latent variables such as a consensus graph, cluster indicators, or low-dimensional node embeddings. Typical goals include:

Consensus structure identification: Learning a graph or subgraph common to all views, representing shared relationships or structure among nodes (Karaaslanli et al., 24 Jan 2024).
Complementary information fusion: Exploiting distinctive structures present only in some views, enabling richer representation or task performance (e.g., clustering, classification) (Jiang et al., 2021).
Node- or edge-level constraints: Imposing sparsity, low-rankness, or structured regularization to extract interpretable or application-specific graph features, such as co-hub nodes shared across layers (Banerjee et al., 13 Dec 2025).
Downstream tasks: Multi-view node classification, link prediction, clustering, and semi-supervised learning benefit from joint graph construction (Chen et al., 2022, Huang et al., 2021).

Formal objective functions vary, but prevalent themes include:

Component	Example Expression (Laplacian Form)
View-wise smoothness	$\mathrm{tr}((X^{(v)})^\top L^{(v)} X^{(v)})$
View consensus (pairwise/fused)	$\sum_{i,j} \gamma_{ij}\\|L^{(i)}-L^{(j)}\\|_p$ or $\sum_v \\|L^{(v)}-L^c\\|_p$ (Karaaslanli et al., 24 Jan 2024)
Graph-structure regularization	$\\|L^{(v)}\\|_{S_p}$ (Schatten- $p$ norm), $\\|L^{(v)}\\|_F^2$
Consensus-graph learning	Minimize $c(\{L^{(v)}-L\}_{v=1}^N)$ for some fusion penalty (Karaaslanli et al., 24 Jan 2024)
Node/hub enforcement	$\\|\mathbf{V}\\|_{2,1}$ for co-hub-node structures (Banerjee et al., 13 Dec 2025)
Rank/cluster enforcement	$\text{rank}(L) = n - k$ or spectral constraints (Ky-Fan variant)

These models are typically subject to Laplacian constraints ( $L1=0$ , symmetry, negative off-diagonal), and sometimes explicit normalization (trace constraints) for identifiability and scale fixing.

2. Consensus, Complementarity, and Regularization Schemes

A central design axis is how to enforce and exploit commonality and complementarity among graph views:

Pairwise regularization: Early models penalize differences between all pairs $(L^{(i)}, L^{(j)})$ , promoting global similarity but lacking explicit modeling of shared structure (Karaaslanli et al., 24 Jan 2024, Liang et al., 2020).
Consensus-graph regularization: More recent models introduce a learned consensus Laplacian $L$ , penalizing each $L^{(v)}$ 's distance to $L$ via fused ( $\ell_{1,1}$ ) or group ( $\ell_{2,1}$ ) norms, allowing direct identification of shared edges and structure (Karaaslanli et al., 24 Jan 2024).
Node-based regularization: To model shared hubs (nodes with high degree across views), block-sparsity/group norms are applied to connection matrices, enforcing common node-centric patterns (Banerjee et al., 13 Dec 2025).
Low-rank/tensor coupling: For scalable settings, the Schatten- $p$ norm is imposed on bipartite or anchor-based multi-view graph tensors, promoting low-rankness and consensus in a computationally tractable way (Jiang et al., 2021, Gao et al., 2021).

The choice of regularizer and consensus mechanism has direct impacts on the interpretability and specificity of the learned graph(s), as well as suitability for particular domains.

3. Computational Algorithms

Optimization in multiview graph learning typically alternates between updates of graph variables and latent variables such as cluster assignments, representations, or anchor weights. Representative approaches include:

Block coordinate minimization: Alternating updates for Laplacians, embeddings, consensus graphs, and auxiliary variables; convex or non-convex subproblems often admit closed-form or efficient solvers (Karaaslanli et al., 24 Jan 2024, Gurugubelli et al., 2020).
Augmented Lagrangian / ADMM: For non-smooth or compositional objectives (e.g., $\ell_1/\ell_{2,1}$ sparsity), multi-block ADMM is employed for convergence and scalability (Banerjee et al., 13 Dec 2025, Karaaslanli et al., 24 Jan 2024).
Anchor/bipartite graph learning: To scale beyond $O(n^2)$ , anchor (bipartite) representations reduce the graph parameterization, with simplex or QP updates for each node-anchor assignment (Jiang et al., 2021, Gao et al., 2021, Fang et al., 2022).
Grassmannian subspace fusion: When merging graphs learned from different GSL methods, subspace geometry (projection distance) is leveraged for fusion prior to downstream GNN training (Ghiasi et al., 2022).
Differentiable graph selection: For GCNs, graph structures are pruned or reweighted by soft/differentiable node selection and shrinkage mechanisms to enable end-to-end learning (Chen et al., 2022, Chen et al., 2022).

Complexity varies by method, with anchor-based and bipartite approaches typically affording linear scaling in $n$ for fixed anchor number $m \ll n$ .

4. Theoretical Guarantees

Recent advances provide identifiability and statistical guarantees for certain classes of multiview graph learning models:

Identifiability: For co-hub-node models, precise conditions are established under which the decomposition of Laplacians into view-specific and shared-hub terms is unique outside of the shared support (Banerjee et al., 13 Dec 2025).
Estimation error bounds: For sub-Gaussian data, the convergence rate in Frobenius norm of the recovered graphs is bounded in terms of the minimal number of samples per view, sparsity level (number of co-hubs), and regularization strengths (Banerjee et al., 13 Dec 2025).
Robustness analysis: Explicit modeling of consistency and inconsistency provides robustness to noisy or partially mismatched views, and alternating minimization schemes converge to KKT points under standard assumptions (Liang et al., 2020).
Expressivity: Heterogeneous message passing in graph-tuple architectures provably exceeds the expressivity of single-graph (commuting operator) architectures, with formal oracle risk dominance (Chen et al., 11 Oct 2025).

5. Applications and Empirical Results

Multiview graph learning models are validated in a diverse range of domains:

Clustering and segmentation: Anchor-based and low-rank tensor methods directly yield cluster labels without post-processing; state-of-the-art accuracy is consistently reported on image, text, and video datasets with 2–6 views and up to tens of thousands of samples (Jiang et al., 2021, Gao et al., 2021, Fang et al., 2022).
Functional connectivity and neuroscience: Consensus and co-hub models reveal group-level and individual structures in brain networks (EEG, fMRI), with identification of default mode network hubs and brain community structure (Banerjee et al., 13 Dec 2025, Karaaslanli et al., 24 Jan 2024).
Robust feature selection: Joint feature selection and graph learning with consensus enforcement improves downstream clustering and interpretable feature subset discovery (Wang et al., 2021, Fang et al., 2022).
Graph neural networks: End-to-end models fuse feature and graph structures with differentiable or learnable weighting, achieving superior semi-supervised accuracy, especially at low label rates (Chen et al., 2022, Chen et al., 2022, Huang et al., 2021).
Scalability: On large datasets ( $n>10^4$ ), anchor/bipartite approaches, together with low-rank regularization, maintain linear or near-linear time complexity (Jiang et al., 2021, Gao et al., 2021, Fang et al., 2022).
Expressive architectures: Non-commuting operator approaches and multi-view graph convolution frameworks achieve top performance in molecular property prediction and cosmological inference (Chen et al., 11 Oct 2025, Adaloglou et al., 2020).

Quantitative results from principal works are summarized in the table below:

Method / Paper	Domain/task	Size/scale	Accuracy/NMI / Metric	Key Feature
CH-MVGL (Banerjee et al., 13 Dec 2025)	fMRI, synthetic	$n\approx 360–1000$	F1 edge recovery > baselines	Co-hub node structure
MGL (Jiang et al., 2021)	Large-scale clustering	$n=4000$ up to $>10^4$	ACC=0.933 (MSRC-v5), NMI=0.893	Anchor, tensor p-norm
MGCN-DNS (Chen et al., 2022)	Semi-supervised GCN	$n=500–9,000$	ACC: +5–15% over best SOTA	Differentiable node selection
MV-GSL (Ghiasi et al., 2022)	Node classification	Cora/Citeseer	ACC: 84.9%, 74.0%	Grassmann fusion
UDBGL (Fang et al., 2022)	Clustering	$n$ up to $60,000$	NMI: 43.6% (avg. across sets)	Anchor, direct labeling

6. Extensions and Open Challenges

Key directions for future research include:

Hybrid structure models: Integration of shared-node (hub, community), edge, and low-rank structures for rich multitask settings; relaxing the global hub-sharing assumption to allow for both shared and private hubs across views (Banerjee et al., 13 Dec 2025).
End-to-end deep multiview models: Further incorporating differentiable graph learning layers in GNNs with automatic weighting/adaptation of each view (Chen et al., 2022, Chen et al., 2022).
Consensus graph learning in dynamic/missing data: Handling incomplete alignment, missing views, and evolving multisource graphs remains an open challenge (Wang et al., 2022).
Efficient solvers: Development of stochastic or randomized approaches for full $O(n^3)$ Laplacian updates, as well as incremental or streaming methods for real-time applications (Banerjee et al., 13 Dec 2025).
Theory and identifiability: Refining error bounds and identifiability theory for high-dimensional, limited-sample, and adversarially noisy settings (Banerjee et al., 13 Dec 2025, Karaaslanli et al., 24 Jan 2024).

7. Comparative Perspectives

Multiview graph learning substantially expands the scope and power of classical (single-view) graph learning and spectral embedding by explicitly modeling both commonality and diversity across views. Consensus-graph and group-structured regularization offer interpretability and direct control of shared structure, while scalable algorithms and geometric fusion mechanisms enable robust integration across modalities and methods. These developments make multiview graph learning essential for contemporary machine learning tasks involving heterogeneous, high-dimensional, or multi-modal data (Karaaslanli et al., 24 Jan 2024, Banerjee et al., 13 Dec 2025, Jiang et al., 2021, Chen et al., 2022).