Laplacian
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Related Articles from SNS
Towards Stable, Globally Expressive Graph Representations with Laplacian Eigenvectors
arXiv:2410.09737v2 Announce Type: replace Abstract: A popular way to improve the expressive power of graph neural networks (GNNs) is to use Laplacian eigenvectors as additional node features, since they can serve both as structural identifiers and global coordinates of nodes. Properly handling the orthogonal group symmetry among eigenvectors is crucial for the stability and generalizability of Laplacian eigenvector augmented GNNs. Previous studies have shown that using a naive $O(p)$-group...
A Nonlocal $p$-Laplacian Interface Model with Sharp Interface
Announce Type: cross Abstract: We propose an energy-based nonlocal $p$-Laplacian interface problem. Neumann interface conditions are naturally formulated via the energy, while Dirichlet conditions are enforced through a penalty term. A key feature is that the model retains a sharp interface, which facilitates extension to other interface problems; we illustrate this by developing a nonlocal approximation for the $p$-Laplacian interface problem with membrane conditions.
Laplacian Representations for Decision-Time Planning
arXiv:2602.05031v2 Announce Type: replace Abstract: Planning with a learned model remains a key challenge in model-based reinforcement learning (RL). In decision-time planning, state representations are critical as they must support local cost computation while preserving long-horizon structure. In this paper, we show that the Laplacian representation provides an effective latent space for planning by capturing state-space distances at multiple time scales.
Adjacency Spectral Radius Under Laplacian Sparsification: Deterministic and Probabilistic Bounds
arXiv:2606.07459v1 Announce Type: cross Abstract: Spielman-Srivastava spectral sparsification preserves Laplacian quadratic forms to within (1 +/- epsilon), but does not directly control the adjacency spectral radius lambda_1, which governs the NIMFA epidemic threshold and arises in spectral clustering. We prove |lambda_1(A_H) - lambda_1(A_G)| <= epsilon(2 Delta - lambda_1) deterministically, with a sharp epsilon*lambda_1 bound for reweighting sparsifiers via Perron-Frobenius monotonicity....
Alcmean's: Unsupervised community detection using local Laplacian, automatic detection of the number of centers
arXiv:2606.09100v1 Announce Type: new Abstract: Community detection is a fundamental problem in the analysis of complex networks. It has applications across social, biological, and financial domains. Traditional algorithms such as Louvain, LPA, and modularity optimization often require manual parameter tuning.
LEGS: Laplacian-Enhanced Gaussian Splatting with a Nonlinear Weighted Loss
Announce Type: new Abstract: 3D Gaussian Splatting (3DGS) has become an efficient explicit representation for radiance field reconstruction and real-time novel view synthesis. However, its standard photometric loss treats flat and structure-rich regions similarly, which may limit the recovery of sharp contours and fine details. Edge-Guided Gaussian Splatting (EGGS) improves structure awareness through edge-guided weighting, but mainly relies on first-order gradient responses and linear...
Resolving the viscosity operator ambiguity on Riemannian manifolds via a kinematic selection principle
arXiv:2605.17502v2 Announce Type: replace-cross Abstract: On a general Riemannian manifold the Navier-Stokes equations admit several inequivalent formulations, differing in the choice of viscous operator: the Hodge Laplacian, the Bochner Laplacian, or the deformation Laplacian. We show that a Lagrangian kinematic construction, in which the strain rate is built from the rate of change of inner products of Lie-dragged connecting vectors, uniquely selects the deformation Laplacian for fluids...
Sinkhorn Normalization of Diffusion Kernels
arXiv:2507.06161v2 Announce Type: replace Abstract: Smoothing a signal based on local neighborhoods is a core operation in machine learning and geometry processing. On well-structured domains such as vector spaces and manifolds, the Laplace operator derived from differential geometry offers a principled approach to smoothing via heat diffusion, with strong theoretical guarantees. However, constructing such Laplacians requires a carefully defined domain structure, which is not always available.
Exponential thermalisation of viscous fluids on negatively curved manifolds
arXiv:2606.02286v1 Announce Type: cross Abstract: The deterministic incompressible Navier-Stokes equations are physically incomplete: any viscous fluid at finite temperature must exhibit thermal fluctuations whose form is dictated by the fluctuation-dissipation relation. We formulate the stochastic Navier-Stokes equations with the kinematically selected deformation Laplacian on compact Riemannian manifolds with strictly negative Ricci curvature. The fluctuation-dissipation relation, derived...
Global exponential stability for the three-dimensional Navier-Stokes equations on hyperbolic space
Announce Type: replace-cross Abstract: We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space $\HH^3$ admit a unique global mild solution for sufficiently small initial data in $L^3(\HH^3)$, and that this solution decays exponentially to zero. The exponential decay rate is $\mu\lambda_\Def^{(3)}$, where $\mu$ is the dynamic viscosity and $\lambda_\Def^{(3)} = 26/9$ is the effective spectral gap of the deformation...