2008
Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems
Coifman R, Kevrekidis I, Lafon S, Maggioni M, Nadler B. Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems. Multiscale Modeling And Simulation 2008, 7: 842-864. DOI: 10.1137/070696325.Peer-Reviewed Original ResearchStochastic systemsEffective free energy surfaceHigh-dimensional stochastic systemsLow-dimensional representationDimensional stochastic systemsRandom walk matrixHigh-dimensional systemsDiffusion maps spaceDimensional representationDiffusion mapsMean squared error criterionReduction coordinatesSquared error criterionComputational physicsDiffusion operatorDimensional systemsFinite differencesRestriction operatorOriginal spaceComputational experimentsFree energy surfaceError criterionEigenfunctionsSimulation runsDifferent simulation runs
2006
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
Coifman R, Lafon S. Diffusion maps. Applied And Computational Harmonic Analysis 2006, 21: 5-30. DOI: 10.1016/j.acha.2006.04.006.Peer-Reviewed Original ResearchMarkov matrixSpectral graph theoryDiffusion mapsGraph theoryMultiscale geometryGeometric descriptionGeometric counterpartMarkov processComplex geometric structuresData parametrizationGeometric structureEfficient representationDiffusion processDimensionality reductionSpectral propertiesData setsEigenfunctionsMachine learningMatrixParametrizationCoordinatesGeometryTheoryVariety of contextsFramework
1993
The Inverse Spectral Method on the Plane
Coifman R, Fokas A. The Inverse Spectral Method on the Plane. Springer Series In Nonlinear Dynamics 1993, 58-85. DOI: 10.1007/978-3-642-58045-1_5.Peer-Reviewed Original Research