Multimodal Graph Laplacians

• Multimodal graph Laplacians are advanced operators that extend classical Laplacians by incorporating multiple edge sets, heterogeneous node types, and higher-order structures.
• They enable joint spectral analysis through methods like joint diagonalization and closest commuting operators, fostering robust clustering and diffusion processes.
• They underpin applications in manifold learning, hypergraph analysis, and scalable optimization, enhancing data representation across complex networked domains.

Multimodal graph Laplacians generalize the classical notion of the Laplacian operator on graphs to contexts where multiple modalities, structures, or interaction types must be simultaneously encoded and analyzed. These modalities may correspond to different edge sets (“layers”), heterogeneous node types, higher-order structures (such as hyperedges or simplices), manifold-valued data, or even composite metrics and inner product spaces. Research in this area spans noncommutative geometry, spectral theory, kernel and multiscale learning, hypergraph theory, and optimization frameworks—all aiming to furnish discrete operators that robustly capture multi-level or multi-source graph information.

1. Foundations: Classical, Edge, and Multimodal Laplacians

The basic graph Laplacian for a simple undirected graph is $L = D - W$, where $D$ is the diagonal degree matrix and $W$ is the weighted adjacency matrix; its spectrum and eigenfunctions support a variety of tasks in clustering, signal analysis, and network science. Moving beyond the vertex level, the edge Laplacian is constructed via noncommutative differential geometry on graphs: vertices form a function algebra while oriented edges generate the module of 1-forms. A canonical Laplace–Beltrami operator acts on edge functions (“1-forms”), with Euclidean metric and bimodule connections encoding discrete geometry (Majid, 2010). The action on a basis element $\omega_{x\to y}$ is: $$\Delta \omega_{x\to y} = \deg(x)\,\omega_{x\to y} - 2 \sum_{z:\, y\to z}\omega_{y\to z} + \sum_{z:\, x\to z}\omega_{x\to z}$$ The spectral theorem gives

$$\operatorname{Spec}(\Delta) = 2\,\operatorname{Spec}(L) \cup \Big\{ \deg(x) \;:\; x\in V,\ \text{mult. } \deg(x)-1 \Big\}$$

This construction underpins multimodal Laplacians by showing how Laplace-type operators act on multiple levels (vertices, edges, etc.), exposing richer graph variations and interactions.

2. Multi-Layer, Manifold, and Hypergraph Extensions

Multimodal graphs often have multiple layers—distinct edge sets between the same vertices, obtained from complementary data sources. Spectral approaches fuse these layers by operating on the spectra (eigenvectors/eigenvalues) of the Laplacians corresponding to each modality or layer. Two methods are prominent:

• Joint matrix factorization: seeks a common eigenbasis $P$ such that $L^{(i)}_{\text{rw}} \approx P \Lambda^{(i)} P^{-1}$, for $M$ layers $i$; the “joint spectrum” $P$ embeds vertices for clustering (Dong et al., 2011).
• Spectral regularization: starts with one layer’s eigenvectors and refines them by enforcing smoothness with respect to another layer, solving

$$\min_{f} \ \frac{1}{2} \| f - u \|^2 + \lambda f^{\top} L_{\text{sym}} f$$

yielding joint eigenvectors $f^*_i = \mu(L_\text{sym} + \mu I)^{-1} u_i$.

For higher-order interactions, hypergraph Laplacians use incidence matrices between simplices to model diffusion not just between vertices but among higher-dimensional structures. The block Laplacian $𝒟_H = [ 𝒟_{p,q} ]_{p,q=0}^n$ encodes diffusion flows for vertices, edges, triangles, etc.; this generalizes classical Laplacians and models group-level contagion, influence, or classification (Aktas et al., 2021). Recent work extends Laplacians to manifold-valued hypergraphs, defining $p$-Laplacians on tangent bundles over Fréchet or pairwise means, capable of capturing complex non-Euclidean signals (e.g., on spheres or SPD matrices) and yielding new equilibrium behaviors in diffusion (Stokke et al., 14 Jul 2025).

3. Joint Diagonalization and Commuting Operator Frameworks

Combining modalities necessitates consistent spectral representations. Joint approximate diagonalization seeks a single orthonormal basis $\bar{V}$ minimizing the total off-diagonality over $m$ Laplacians: $$\min_{\bar{V}: \bar{V}^\top \bar{V}=I} \sum_{i=1}^m \| \bar{V}^\top L_i \bar{V} - \mathrm{diag}( \bar{V}^\top L_i \bar{V}) \|_F^2$$ (see JADE algorithm (Eynard et al., 2012)). This basis enables fused diffusion maps, robust spectral clustering, and manifolds that accurately reflect shared intrinsic structure.

Alternatively, the closest commuting operators (CCO) formulation modifies Laplacians $L_1, L_2$ minimally so that $[\widetilde{L}_1, \widetilde{L}_2] = 0$ (commute), allowing for joint diagonalization. The optimization is: $$\min_{\widetilde{L}_k} \sum_{k=1,2} \| \widetilde{L}_k - L_k \|_F^2 \quad \text{s.t.} \quad [\widetilde{L}_1, \widetilde{L}_2] = 0$$ yielding a joint eigenbasis for diffusion operators, facilitating dimensionality reduction and clustering that consistently reflects all modalities (Bronstein et al., 2013).

4. Generalized Inner Products, Multiscale, and Learning Approaches

The inner product Laplacian framework provides a formal generalization: by specifying arbitrary positive-definite inner product matrices on vertex and edge spaces ($M_V = Q_V^2$, $M_E = Q_E^2$), one constructs Laplacians that encode both combinatorial and domain-specific information. The Hodge-type formula is: $$IPL[i] = Q_i \cdot (_i)^* \cdot Q_{i-1}^{-2} \cdot _i \cdot Q_i + Q_i^{-1} \cdot _{i+1} \cdot Q_{i+1}^2 \cdot (_{i+1})^* \cdot Q_i^{-1}$$ Special cases recover classical, normalized, directed, and hypergraph Laplacians; explicit conformality parameters quantify the effect of inner product choice on key spectral bounds, including Cheeger and expander mixing inequalities. This modality enables fusion of heterogeneous graph data, direct incorporation of side information, and tailored spectral analysis (Aksoy et al., 14 Apr 2025).

Multiscale Laplacians combine graph Laplacian operators at several scales: $$L_{\text{multiscale}} = \sum_{t=0}^m c_t \cdot L_t^{p_t}$$ where $L_t$ represent Laplacians built from kernels using Hermite polynomials at different scales $\sigma_t$ (Merkurjev et al., 2021). These methods—applied in manifold regularization and MBO diffusion—show improved performance in classification and semi-supervised learning with limited labeled data, utilizing spectral projection and implicit propagation for stability.

Recent graph Laplacian learning algorithms optimize both graph structure and smooth signal representation by alternating updates:

• Graph Laplacian $L$ inferred to minimize $x^\top L x$ (“energy”) under Laplacian constraints.
• Data representation $Y$ denoised via $Y = (I + \alpha L)^{-1} X$. Such schemes—enforced by a Gaussian prior on latent variables—recover topologies consistent with underlying relationships in synthetic and real-world settings (Dong et al., 2014).

Sparse signal models use a graph dictionary: each observation is presumed to be generated on a Laplacian that is a weighted sum of $K$ atoms, $L_t = \sum_{k=1}^K \delta_{tk} L_{w_k}$, and signals $x_t = G(L_t) \Delta$, with a bilinear primal-dual splitting algorithm for MAP estimation over atoms $W$ and activations $\Delta$ (Cappelletti et al., 8 Nov 2024). This approach provides interpretable, task-adaptive graph representations.

5. Spectral Properties and Analytical Testbeds

Analytical solutions for multidimensional grid graphs and their Laplacians provide a valuable substrate for testing and benchmarking multimodal methods (Kłopotek, 2017). Closed-form formulas for eigenvalues and eigenvectors of the combinatorial, normalized, and random walk Laplacians are given, e.g.: $$\lambda_{[z_1,\dots,z_d]} = \sum_{j=1}^d \left( 2\sin\frac{\pi z_j}{2n_j} \right)^2$$ These formulas expose key differences in spectral distributions (non-uniformity, boundary shifts, scaling effects) and cluster simulation capabilities in weighted settings, underscoring the complexity of spectral assumptions in multimodal Laplacian algorithms. Weighted grids enable “soft” cluster structure and sensitivity analysis for multimodal clustering.

6. Algorithmic Frameworks, Scalability, and Real-World Applications

Multimodal Laplacian operators necessitate efficient solvers. Lean Algebraic Multigrid (LAMG) is optimized for linear systems $A x = b$ where $A$ is a graph Laplacian; its setup (node aggregation by affinity, piecewise-constant interpolation, energy correction) and iterative solves scale linearly with the graph’s number of edges (Livne et al., 2011). LAMG can be extended for eigenvalue problems, spectral clustering, and general graph optimization—offering fast convergence across diverse, multimodal graphs.

Applications of multimodal Laplacians include:

• Clustering vertices, especially in social networks or bibliographic datasets characterized by multiple interaction modalities (Dong et al., 2011, Eynard et al., 2012).
• Manifold learning and retrieval, using joint diagonalization to improve embedding consistency and robustness to topological noise (Eynard et al., 2012, Bronstein et al., 2013).
• Signal processing and diffusion modeling, exploiting learned or multiscale Laplacians for denoising, semi-supervised classification, and efficient diffusion on complex graphs (Dong et al., 2014, Merkurjev et al., 2021).
• Hypergraph-based contagion, influence, and classification tasks where higher-order interactions are essential (Aktas et al., 2021, Stokke et al., 14 Jul 2025).
• Sparse dictionary models for multivariate (e.g., brain EEG) signals that improve interpretability and classification accuracy (Cappelletti et al., 8 Nov 2024).

7. Challenges and Future Directions

While multimodal Laplacian frameworks substantially extend the modeling and analytical power of classical spectral graph theory, several open questions remain:

• Theoretical guarantees on spectral convergence, especially when applying secondary measures (e.g., SNN graphs have same Laplacian limit as $k$-NN graphs (Neuman, 2023)).
• Designing inner product matrices and aggregation strategies that exploit domain knowledge without loss of analytic tractability or computational scalability.
• Handling large, heterogeneous, multilayer and manifold-valued hypergraphs—requiring development of scalable optimization and spectral algorithms.
• Extending robustness and interpretability in dictionary signal models and learning frameworks to dynamic or streaming multimodal graphs.
• Further paper of equilibrium and limiting behaviors in diffusion induced by hypergraph Laplacians, especially in non-Euclidean settings or with nonconstant dynamics (Stokke et al., 14 Jul 2025).
• Exploring generalized boundary conditions and subgraph spectral properties (e.g., Neumann, Dirichlet eigenvalues) via convergent inner product Laplacian sequences (Aksoy et al., 14 Apr 2025).

Advances in multimodal graph Laplacians continue to inform the analysis, modeling, and learning of complex networked data, spanning social, biological, and physical domains, while challenging established frameworks in spectral geometry and graph signal processing.