graph spectral projection

This work addresses the challenge of efficiently computing leading eigenvectors in dynamic graphs, where frequent updates to the adjacency or Laplacian matrix render traditional eigendecomposition methods computationally prohibitive. To overcome this limitation, the authors propose a fast spectral embedding update framework based on Rayleigh-Ritz projection. By leveraging eigenvector perturbation analysis, the method constructs a low-dimensional approximate invariant subspace that preserves high approximation accuracy while substantially reducing computational and memory costs. Experimental results demonstrate that the proposed approach outperforms existing techniques in both the quality of leading eigenvector approximation and performance on downstream tasks—such as influential node identification and node clustering—offering a compelling balance between efficiency and accuracy.

This work addresses the inefficiency of conventional dimensionality reduction in contextual bandits with graph-structured arms, where ignoring graph information leads to exploration complexity scaling with the ambient dimension $d$ rather than the effective dimension $k$. The authors propose projecting arm features onto the low-frequency subspace of the graph Laplacian and running linear UCB in this $k$-dimensional space, establishing the first $\tilde{O}(k\sqrt{T})$ regret bound for spectral projection-based bandits. The analysis reveals that the practical cost of high-frequency reward components depends on their influence along the policy’s trajectory rather than their total energy. A threshold-free subspace spectral comparison criterion is introduced to predict algorithmic performance. Experiments on six real-world datasets show a 15-fold average reduction in cumulative regret, outperforming existing graph-aware methods in five cases, with failures precisely aligning with mismatches between the graph spectral subspace and the reward signal.

Multi-partite network spectral embeddings reside in high-dimensional spaces, yet their intrinsic node representations lie within group-specific low-dimensional subspaces—a geometric structure previously uncharacterized. Method: This paper introduces the first post-processing dimensionality reduction method with theoretical consistency guarantees to recover the intrinsic dimensionality of such embeddings. Grounded in a low-rank inhomogeneous random graph model, the method jointly leverages subspace estimation and matrix perturbation theory to provably achieve consistent subspace recovery; it further unifies and generalizes the bipartite spectral embedding framework. Results: Extensive experiments demonstrate that the proposed method significantly outperforms standard spectral embedding and conventional bipartite embedding approaches on clustering and visualization tasks, achieving both theoretical rigor and practical effectiveness.

Learning the spectral decomposition of the Laplacian operator from unstructured high-dimensional data—such as 3D point clouds or image manifolds—typically requires explicit discretization, mesh construction, or solving eigenvalue problems, limiting scalability and applicability to unknown geometries. Method: We propose an end-to-end framework that jointly learns the implicit spectral basis, eigenvalues, and density-induced metric of the Laplacian directly from raw data—without constructing differential operators, discretizing domains, or solving eigenproblems. Leveraging optimal approximation theory, we parameterize the operator and its spectrum via neural networks, minimizing reconstruction error under a probe function distribution. Contribution/Results: The method is fully unsupervised, mesh-free, dimension-agnostic, and geometry-agnostic. Experiments demonstrate that the learned spectral basis exhibits Laplacian-like properties—yielding interpretable, generalizable, and scalable representations across diverse unstructured datasets, including high-dimensional settings.

Conventional diffusion models for graph generation suffer from O(n²) computational complexity in the node space, hindering scalability. Method: This paper proposes GGSD, the first model to jointly integrate graph Laplacian spectral decomposition with denoising diffusion probabilistic modeling, establishing a novel spectral-space diffusion paradigm. GGSD employs spectral truncation for efficient low-dimensional representation and introduces a linear-complexity, permutation-invariant Transformer architecture that supports node feature fusion. Crucially, it generates graph structures directly in the node space while achieving theoretical O(n) complexity. Contribution/Results: Extensive experiments demonstrate that GGSD significantly outperforms state-of-the-art methods on both synthetic and real-world graph datasets, achieving superior trade-offs among generation speed, structural fidelity, and scalability.

This work addresses the tendency of Laplacian-constrained graphical models—such as the Laplacian-constrained Gaussian graphical model (LCGGM) and the Hüsler–Reiss model—to yield overly dense graphs during structure learning, which compromises interpretability and scalability. The paper introduces, for the first time, spectral graph sparsification as a post-processing step: without requiring additional hyperparameter tuning, it replaces the original Laplacian estimate with a spectrally approximated sparse Laplacian and refits the model. This approach effectively enhances both the sparsity and accuracy of the estimated graph structure. Integrating spectral graph theory, Laplacian-constrained Gaussian graphical models, extreme-value graphical models, and graph sparsification techniques, the method demonstrates superior performance on Erdős–Rényi and stochastic block model simulations and validates its practical utility on real-world data.

This work addresses the high computational cost of computing multiple nonlinear eigenvectors in nonlinear spectral clustering by proposing a direct multiway spectral clustering algorithm based on the p-norm (with p ∈ (1,2]). The algorithm is implemented for the first time within the C++ GraphBLAS framework, unifying its core operations into sparse linear algebraic expressions. By integrating shared-memory parallelism and p-norm-specific optimizations, the approach achieves substantial gains in computational efficiency. Experimental results on a large-scale graph with 8 million nodes and 48 million edges demonstrate excellent strong scaling performance, while the clustering quality surpasses that of existing methods in terms of balanced graph cut metrics.

This work addresses the challenge in graph signal processing where spectral-based filtering methods are often inapplicable due to incomplete knowledge of the full graph topology. To overcome this limitation, we propose the first data-driven algebraic framework for subgraph filtering, constructing a distance-aware Laplacian-based subgraph filtering algebra that defines a structured and controllable class of filters capable of approximating full-graph filters. Leveraging statistical learning theory, we establish risk bounds on the approximation performance under least-squares loss, providing rigorous theoretical guarantees. Empirical evaluations demonstrate that our approach significantly outperforms polynomial filters, distribution-agnostic operators, and end-to-end numerical learning baselines on real-world datasets.

This work extends determinantal point process (DPP) sampling theory to general non-Euclidean metric spaces—such as Riemannian manifolds and weighted graphs—for the first time. By constructing kernel functions based on spectral properties of Laplace and Markov diffusion operators, the authors propose an adaptive DPP sampling framework that automatically adjusts to the intrinsic dimensionality of the data. Leveraging tools from Weyl’s law, Dirichlet forms, and pseudodifferential operator theory, the method achieves a sampling error rate of $O(n^{-1/2 - 1/(2d_{\text{int}})})$ on compact manifolds and k-nearest neighbor graphs, significantly outperforming conventional i.i.d. sampling and matching the optimal rates known in Euclidean settings.

Traditional vector retrieval relies on pairwise geometric similarity, which struggles to simultaneously achieve semantic alignment and consistency with the head-tail distribution of data. This work proposes a Graph Wiring framework combined with Spectral Indexing, modeling the embedding space as an energy network induced by the topology of feature column vectors. By integrating geometric similarity with spectral structural information and introducing τ-modulation for adaptive retrieval, the method leverages spectral graph theory, energy-based modeling, and epiplexity analysis. Implemented using the open-source arrowspace library, it significantly outperforms purely geometric retrieval across multiple benchmarks and industrial applications, effectively enhancing both semantic alignment and distributional consistency to meet the demands of modern RAG systems for flexible and efficient retrieval.