A tutorial on dynamic Laplacian for time-evolving data clusters.
problem Handling data with a time component and lying on a time-evolving manifold.
method Dynamic generalization of the Laplacian, SEBA algorithm for feature separation.
result Clusters correspond to long-lived coherent collections in time-evolving data.
ULES embeds dynamic networks with stability guarantees.
problem Stability of time-varying node embeddings in evolving networks.
method Unfolded Laplacian Spectral Embedding (ULSE) using normalized Laplacian operators.
result ULES satisfies cross-sectional and longitudinal stability under dynamic stochastic block model.
New method clusters evolving networks using spatio-temporal graph Laplacian.
problem Clustering communities in time-varying graphs.
method Extends spectral clustering to dynamic graphs using CCA and spatio-temporal graph Laplacian.
result The spatio-temporal graph Laplacian clearly interprets cluster evolution over time.
Kernel Dynamic Mode Decomposition reconstructs dynamical systems using Laplacian kernel.
problem Reconstructing spatial-temporal dynamics of complex systems.
method Kernel Dynamic Mode Decomposition with Laplacian kernel.
result Laplacian kernel allows for the closability of Koopman operators in RKHS, enabling reconstruction.
LAD detects anomalies in dynamic graphs using Laplacian matrix.
problem Anomaly detection in temporal graphs for real-world applications.
method LAD uses the spectrum of the Laplacian matrix to model graph snapshots and temporal dependencies.
result LAD outperforms state-of-the-art methods in synthetic and real-world datasets.
We prove that torsion-free G_2 structures are (weakly) dynamically stable along the Laplacian flow for closed G_2 structures. More precisely, given a torsion-free G_2 structure φ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to φ will converge to a to…
About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recen…
In this paper we first derive several results concerning the Lp spectrum of arithmetic locally symmetric spaces whose $\Q$-rank equals one. In particular, we show that there is an open subset of $\C$ consisting of eigenvalues of the Lp Laplacian if p<2 and that corresponding eigenfunctions are given by certain…
Koopman operator theory simplifies complex systems analysis.
problem Analyzing nonlinear dynamical systems and complex networks.
method Estimating Koopman operator from data to reveal system properties.
result Koopman operators provide insights into system characteristics.
A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.
problem Ambiguity in viscous operator choice for Navier-Stokes equations on Riemannian manifolds.
method Kinematic construction of strain rate from Lie-dragged vectors, excluding Hodge Laplacian due to antisymmetric part.
result Kinematic selection uniquely identifies the deformation Laplacian, resolving analytical obstructions.
As an important Markov Chain Monte Carlo (MCMC) method, stochastic gradient Langevin dynamics (SGLD) algorithm has achieved great success in Bayesian learning and posterior sampling. However, SGLD typically suffers from slow convergence rate due to its large variance caused by the stochastic gradient. In order to allev…
Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.
problem Investigate semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces.
method Analyze eigenfunctions in low, critical, and high energy regimes using quantum ergodicity and equidistribution.
result Eigenfunctions in different regimes converge to distinct measures: invariant, Liouville, or equidistributed.
Introduces PELP for graph-enhanced word embeddings.
problem Combining graph side-information into static word embeddings.
method Probabilistic embeddings using Laplacian priors.
result Unified and flexible approach to various embedding methods.
Improved spectral clustering guarantees for dynamic stochastic block models.
problem Analyzing Spectral Clustering in dynamic stochastic block models.
method Extending guarantees to sparse and smooth DSBM, linking sparsity and smoothness.
result Improved error bounds for consistent recovery in dynamic DSBM.
The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian)…
End-to-end graph-based SSL learns all graph factors dynamically.
problem Learning quality of graph in SSL is crucial but difficult.
method Proposes an end-to-end approach to optimize all graph factors.
result Demonstrates effectiveness on benchmark datasets.
Distance plays a fundamental role in measuring similarity between objects. Various visualization techniques and learning tasks in statistics and machine learning such as shape matching, classification, dimension reduction and clustering often rely on some distance or similarity measure. It is of tremendous importance t…
Quantum-classical correspondence links graph Laplacians to manifold dynamics.
problem Establishing a quantum-classical link for graph Laplacians on manifolds.
method Using semiclassical pseudodifferential operators and coherent states, the paper connects graph Laplacians to geodesic flows on manifolds.
result The geodesic flow on manifolds can be approximated by matrix dynamics on discrete samples.
Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the s…
We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or h…
New method clusters directed graphs using Koopman operators.
problem Challenges in clustering directed graphs, especially complex eigenvalues and lack of cluster definition.
method Relate graph Laplacians to transfer operators and metastable sets in stochastic systems, derive clustering algorithms for directed and time-evolving graphs.
result Clusters can be interpreted as coherent sets, useful for analyzing transport and mixing processes.
SpecRaGE learns robust multi-view representations using graph Laplacians and neural networks.
problem Challenges in generalizing and scaling multi-view representation learning methods.
method SpecRaGE integrates graph Laplacian methods with neural networks to learn robust representations.
result SpecRaGE outperforms state-of-the-art methods in noisy and contaminated data scenarios.
Single-particle electron cryomicroscopy is an essential tool for high-resolution 3D reconstruction of proteins and other biological macromolecules. An important challenge in cryo-EM is the reconstruction of non-rigid molecules with parts that move and deform. Traditional reconstruction methods fail in these cases, resu…
New algorithm updates eigenvectors of evolving graphs efficiently.
problem Updating eigenvectors of dynamic graphs.
method Subspace projection based on Rayleigh-Ritz projections.
result Strong performance in eigenvector approximation and downstream tasks.
We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. A corollary of this dynamical trace formula is a dynamical resonance-wave trac…
Stylized facts of empirical assets log-returns Z include the existence of (semi) heavy tailed distributions fZ(z) and a non-linear spectrum of Hurst exponents τ(β). Empirical data considered are daily prices of 10 large indices from 01/01/1990 to 12/31/2004. We propose a stylized model of price dynamics which is…
Defines vector Laplacian on statistical manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines and derives vector Laplacian formula.
result Derives formula for vector Laplacian.
The aim of this paper is to study the spectrum of the Lp Laplacian and the dynamics of the Lp heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in t…
EGDL predicts TB outbreaks with deep learning, integrating epidemiological models.
problem Predicting TB outbreaks with complex spatiotemporal dynamics.
method Modified MN-SIR model with Bayesian inference, deep neural networks.
result EGDL delivers robust and accurate TB outbreak predictions.
New method learns under latent group sparsity using network dynamics.
problem Sparse learning under latent group structure without prior group information.
method Heat-flow-based local network dynamics incorporating Laplacian geometry.
result Automatic interpolation between lasso and group lasso penalties.
BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.
problem Comparing combinatorial and Hodge Laplacians for discrete data.
method Introducing Boundary-Induced Graph (BIG) Laplacians using DEC.
result BIG Laplacian eigenvalues converge to Hodge Laplacian for simple shapes.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesi…
In this article, we introduce a new method (based on Perelman's lambda-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (A) a Ricci-flat metric is a local maximizer of lambda in a C^2,alpha-sense iff its Lichne…
Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.
problem Analyzing the Ruelle dynamical zeta function on locally symmetric spaces with flat vector bundles.
method Meromorphic extension and regularisation of the dynamical zeta function, relating it to the complex valued analytic torsion.
result The leading term of the dynamical zeta function at zero is related to the regularised determinant of the flat Laplacian.
Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.
problem Analyzing sectoriality of Laplacian and Lichnerowicz Laplacian on asymptotically hyperbolic spaces.
method Proves sectoriality in weighted Hölder spaces using asymptotically hyperbolic metrics.
result Analytic semigroups apply, yielding well-posedness results for parabolic evolution equations.
The paper extends Laplacian spectra approximations to vector bundles.
problem Approximating the spectrum of the connection Laplacian.
method Extending the graph connection Laplacian to vector bundles and proving spectrum approximation.
result The spectrum of the extended operator approximates the spectrum of the connection Laplacian.
Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.
problem Understanding Laplacian eigenfunctions on complex hyperbolic quotients.
method Combining fractal uncertainty principle and Ratner theory to analyze measure supports.
result Semiclassical measures support is either cosphere bundle or a compact submanifold.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph p-Laplacian. Unlike the …
The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.
problem Eigenvalue comparison theorems for Witten-Laplacian and weighted p-Laplacian on manifolds with modified Ricci curvature. method Established Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted p-Laplacian on geodesic balls. result Successfully set up eigenvalue comparison theorems for the Witten-Laplacian and weighted p-Laplacian. Extended bounds on small eigenvalues for pseudo-Laplacians on hyperbolic surfaces.
problem Bounding small eigenvalues of pseudo-Laplacians on hyperbolic surfaces.
method Extended Otal-Rosas bound and Colin de Verdière's spectral theory to hyperbolic surfaces with multiple cusps.
result Extended bounds on small eigenvalues for pseudo-Laplacians.
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
Study quantum diffusion on spectral triples and spinor bundles.
problem Characterize quantum diffusion on almost commutative spectral triples.
method Spin geometry, C *-Dirichlet forms, quantum stochastic flows.
result Existence of covariant quantum stochastic flows on spinor bundles.
Study Hodge Laplacians for manifold data, improving error bounds.
problem Approximating Laplace-Beltrami operator on differential forms.
method Higher-order graph Laplacians (Hodge Laplacians) as approximations.
result High-probability error bound for Dirichlet forms.
Root Laplacian Eigenmaps help in spectral embedding of graphs.
problem Efficient spectral embedding of graphs.
method Square root of graph-Laplacian operator.
result Improved spectral embedding techniques.
New perspective on G2-structures flow from DeTurck Laplacian.
problem Understanding G2-structures and their flows.
method Introducing a new flow (DeTurck Laplacian flow) for G2-structures.
result DeTurck Laplacian flow is a flow of G2-structures.
Explains BV Laplacian on half-densities in simple terms.
problem None explicitly stated; focuses on explanation.
method Didactical review of BV Laplacian on half-densities.
result Explains BV Laplacian concept in plain language.