Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions
Coifman R, Lafon S. Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions. Applied And Computational Harmonic Analysis 2006, 21: 31-52. DOI: 10.1016/j.acha.2005.07.005.Peer-Reviewed Original ResearchEntire space RnProlate spheroidal wave functionsLaplace-Beltrami operatorSpheroidal wave functionsFunction fSubmanifold of RnNyström methodSpace RnFourier modesSample extensionGeometric harmonicsEmpirical functionWave functionsSimple schemeExtension schemeLarge domainsSpecific familySchemeRnIntrinsic frequency spectrumExtensionFrequency spectrumSubmanifoldsEigenfunctionsSlepianDiffusion maps, spectral clustering and reaction coordinates of dynamical systems
Nadler B, Lafon S, Coifman R, Kevrekidis I. Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Applied And Computational Harmonic Analysis 2006, 21: 113-127. DOI: 10.1016/j.acha.2005.07.004.Peer-Reviewed Original ResearchFokker-Planck operatorDynamical systemsDifferential operatorsHigh-dimensional stochastic systemsRandom walkProbability distributionDimensional stochastic systemsStochastic differential equationsCorresponding differential operatorComplex dynamical systemsTime evolutionLong-time asymptoticsLow-dimensional Euclidean spaceGeneral probability distributionNormalized graph LaplacianLaplace-Beltrami operatorDimensional Euclidean spaceDiffusion mapsLong-time evolutionSpectral clusteringStochastic systemsDifferential equationsHigh-dimensional dataSlow variablesLarge-scale simulations