## 2021 |

Felix X.-F. Ye Sichen Yang, Mauro Maggioni Nonlinear model reduction for slow-fast stochastic systems near manifolds Journal Article 2021. Abstract | Links | BibTeX | Tags: inverse problems, Machine learning, Manifold Learning, model reduction, random walks, statistics, stochastic systems @article{YYM:ATLAS2, title = {Nonlinear model reduction for slow-fast stochastic systems near manifolds}, author = {Felix X.-F. Ye, Sichen Yang, Mauro Maggioni}, url = {https://arxiv.org/abs/2104.02120v1}, year = {2021}, date = {2021-04-05}, abstract = {We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we estimate the invariant manifold, a process of the effective (stochastic) dynamics on it, and construct an efficient simulator thereof. These estimation steps can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.}, keywords = {inverse problems, Machine learning, Manifold Learning, model reduction, random walks, statistics, stochastic systems}, pubstate = {published}, tppubtype = {article} } We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we estimate the invariant manifold, a process of the effective (stochastic) dynamics on it, and construct an efficient simulator thereof. These estimation steps can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them. |

## 2011 |

Lee, J; Maggioni, Mauro Multiscale Analysis of Time Series of Graphs Conference Proc. SampTA, 2011. BibTeX | Tags: random walks, spectral graph theory @conference{LM:sampta11, title = {Multiscale Analysis of Time Series of Graphs}, author = {J Lee and Mauro Maggioni}, year = {2011}, date = {2011-01-01}, booktitle = {Proc. SampTA}, keywords = {random walks, spectral graph theory}, pubstate = {published}, tppubtype = {conference} } |

## 2010 |

Jones, Peter W; Maggioni, Mauro; Schul, Raanan Universal local manifold parametrizations via heat kernels and eigenfunctions of the Laplacian Journal Article Ann. Acad. Scient. Fen., 35 , pp. 1–44, 2010, (http://arxiv.org/abs/0709.1975). BibTeX | Tags: diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory @article{jms:UniformizationEigenfunctions2, title = {Universal local manifold parametrizations via heat kernels and eigenfunctions of the Laplacian}, author = {Peter W Jones and Mauro Maggioni and Raanan Schul}, year = {2010}, date = {2010-01-01}, journal = {Ann. Acad. Scient. Fen.}, volume = {35}, pages = {1–44}, note = {http://arxiv.org/abs/0709.1975}, keywords = {diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory}, pubstate = {published}, tppubtype = {article} } |

## 2008 |

Coifman, Ronald R; Maggioni, Mauro Geometry Analysis and Signal Processing on Digital Data, Emergent Structures, and Knowledge Building Miscellaneous SIAM News, 2008. BibTeX | Tags: diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory @misc{CM:SiamNews, title = {Geometry Analysis and Signal Processing on Digital Data, Emergent Structures, and Knowledge Building}, author = {Ronald R Coifman and Mauro Maggioni}, year = {2008}, date = {2008-11-01}, howpublished = {SIAM News}, keywords = {diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory}, pubstate = {published}, tppubtype = {misc} } |

Szlam, Arthur D; Maggioni, Mauro; Coifman, Ronald R Regularization on Graphs with Function-adapted Diffusion Processes Journal Article Jour. Mach. Learn. Res., (9), pp. 1711–1739, 2008, ((YALE/DCS/TR1365, Yale Univ, July 2006)). BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, random walks, semisupervised learning, spectral graph theory @article{SMC:GeneralFrameworkAdaptiveRegularization, title = {Regularization on Graphs with Function-adapted Diffusion Processes}, author = {Arthur D Szlam and Mauro Maggioni and Ronald R Coifman}, year = {2008}, date = {2008-08-01}, journal = {Jour. Mach. Learn. Res.}, number = {9}, pages = {1711–1739}, note = {(YALE/DCS/TR1365, Yale Univ, July 2006)}, keywords = {diffusion geometry, Machine learning, Manifold Learning, random walks, semisupervised learning, spectral graph theory}, pubstate = {published}, tppubtype = {article} } |

Jones, Peter W; Maggioni, Mauro; Schul, Raanan Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels Journal Article Proc. Nat. Acad. Sci., 105 (6), pp. 1803–1808, 2008. BibTeX | Tags: diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory @article{jms:UniformizationEigenfunctions, title = {Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels}, author = {Peter W Jones and Mauro Maggioni and Raanan Schul}, year = {2008}, date = {2008-02-01}, journal = {Proc. Nat. Acad. Sci.}, volume = {105}, number = {6}, pages = {1803–1808}, keywords = {diffusion geometry, heat kernels, Laplacian eigenfunctions, Manifold Learning, multiscale analysis, random walks, spectral graph theory}, pubstate = {published}, tppubtype = {article} } |

## 2007 |

Coifman, Ronald R; Maggioni, Mauro Multiscale Data Analysis with Diffusion Wavelets Journal Article Proc. SIAM Bioinf. Workshop, Minneapolis, 2007. BibTeX | Tags: diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems @article{CM:MsDataDiffWavelets, title = {Multiscale Data Analysis with Diffusion Wavelets}, author = {Ronald R Coifman and Mauro Maggioni}, year = {2007}, date = {2007-04-01}, journal = {Proc. SIAM Bioinf. Workshop, Minneapolis}, keywords = {diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

Mahadevan, Sridhar; Maggioni, Mauro Proto-value Functions: A Spectral Framework for Solving Markov Decision Processes Journal Article JMLR, 8 , pp. 2169–2231, 2007. BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory @article{smmm:jmrl1, title = {Proto-value Functions: A Spectral Framework for Solving Markov Decision Processes}, author = {Sridhar Mahadevan and Mauro Maggioni}, year = {2007}, date = {2007-01-01}, journal = {JMLR}, volume = {8}, pages = {2169–2231}, keywords = {diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory}, pubstate = {published}, tppubtype = {article} } |

## 2006 |

Coifman, Ronald R; Maggioni, Mauro Diffusion Wavelets Journal Article Appl. Comp. Harm. Anal., 21 (1), pp. 53–94, 2006, ((Tech. Rep. YALE/DCS/TR-1303, Yale Univ., Sep. 2004)). BibTeX | Tags: diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems @article{CMDiffusionWavelets, title = {Diffusion Wavelets}, author = {Ronald R Coifman and Mauro Maggioni}, year = {2006}, date = {2006-07-01}, journal = {Appl. Comp. Harm. Anal.}, volume = {21}, number = {1}, pages = {53–94}, note = {(Tech. Rep. YALE/DCS/TR-1303, Yale Univ., Sep. 2004)}, keywords = {diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

Bremer, James Jr. C; Coifman, Ronald R; Maggioni, Mauro; Szlam, Arthur D Diffusion Wavelet Packets Journal Article Appl. Comp. Harm. Anal., 21 (1), pp. 95–112, 2006, ((Tech. Rep. YALE/DCS/TR-1304, 2004)). BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems @article{DiffusionWaveletPackets, title = {Diffusion Wavelet Packets}, author = {James Jr. C Bremer and Ronald R Coifman and Mauro Maggioni and Arthur D Szlam}, year = {2006}, date = {2006-07-01}, journal = {Appl. Comp. Harm. Anal.}, volume = {21}, number = {1}, pages = {95–112}, note = {(Tech. Rep. YALE/DCS/TR-1304, 2004)}, keywords = {diffusion geometry, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

Coifman, Ronald R; Lafon, Stephane; Maggioni, Mauro; Keller, Y; Szlam, A D; Warner, F J; Zucker, S W Geometries of sensor outputs, inference, and information processing Inproceedings Athale, John Zolper; Eds. Intelligent Integrated Microsystems; Ravindra C A (Ed.): Proc. SPIE, pp. 623209, 2006. BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems @inproceedings{CLMKSWZ:GeometrySensorOutputs, title = {Geometries of sensor outputs, inference, and information processing}, author = {Ronald R Coifman and Stephane Lafon and Mauro Maggioni and Y Keller and A D Szlam and F J Warner and S W Zucker}, editor = {John Zolper; Eds. C Intelligent Integrated Microsystems; Ravindra A. Athale}, year = {2006}, date = {2006-05-01}, booktitle = {Proc. SPIE}, volume = {6232}, pages = {623209}, keywords = {diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {inproceedings} } |

Maggioni, Mauro; Mahadevan, Sridhar Fast Direct Policy Evaluation using Multiscale Analysis of Markov Diffusion Processes Inproceedings ICML 2006, pp. 601–608, 2006. BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory @inproceedings{smmm:FastDirectMDP, title = {Fast Direct Policy Evaluation using Multiscale Analysis of Markov Diffusion Processes}, author = {Mauro Maggioni and Sridhar Mahadevan}, year = {2006}, date = {2006-01-01}, booktitle = {ICML 2006}, pages = {601–608}, keywords = {diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory}, pubstate = {published}, tppubtype = {inproceedings} } |

Mahadevan, Sridhar; Ferguson, Kim; Osentoski, Sarah; Maggioni, Mauro Simultaneous Learning of Representation and Control In Continuous Domains Inproceedings AAAI, AAAI Press, 2006. BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory @inproceedings{smkfsomm:SimLearningReprControlContinuous, title = {Simultaneous Learning of Representation and Control In Continuous Domains}, author = {Sridhar Mahadevan and Kim Ferguson and Sarah Osentoski and Mauro Maggioni}, year = {2006}, date = {2006-01-01}, booktitle = {AAAI}, publisher = {AAAI Press}, keywords = {diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory}, pubstate = {published}, tppubtype = {inproceedings} } |

## 2005 |

Coifman, Ronald R; Maggioni, Mauro; Zucker, Steven W; Kevrekidis, Ioannis G Geometric diffusions for the analysis of data from sensor networks Journal Article Curr Opin Neurobiol, 15 (5), pp. 576–84, 2005. BibTeX | Tags: diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems @article{CMZK:CONB, title = {Geometric diffusions for the analysis of data from sensor networks}, author = {Ronald R Coifman and Mauro Maggioni and Steven W Zucker and Ioannis G Kevrekidis}, year = {2005}, date = {2005-10-01}, journal = {Curr Opin Neurobiol}, volume = {15}, number = {5}, pages = {576–84}, keywords = {diffusion geometry, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

Coifman, Ronald R; Lafon, Stephane; Lee, Ann B; Maggioni, Mauro; Nadler, B; Warner, Frederick; Zucker, Steven W Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps Journal Article Proceedings of the National Academy of Sciences of the United States of America, 102 (21), pp. 7426-7431, 2005. BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems @article{DiffusionPNAS, title = {Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps}, author = {Ronald R Coifman and Stephane Lafon and Ann B Lee and Mauro Maggioni and B Nadler and Frederick Warner and Steven W Zucker}, year = {2005}, date = {2005-01-01}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, volume = {102}, number = {21}, pages = {7426-7431}, keywords = {diffusion geometry, Machine learning, Manifold Learning, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

Coifman, Ronald R; Lafon, S; Lee, A B; Maggioni, Mauro; Nadler, B; Warner, Frederick; Zucker, Steven W Geometric diffusions as a tool for harmonic analysis and structure definition of data: Multiscale methods Journal Article Proceedings of the National Academy of Sciences of the United States of America, 102 (21), pp. 7432–7438, 2005. BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems @article{DiffusionPNAS2, title = {Geometric diffusions as a tool for harmonic analysis and structure definition of data: Multiscale methods}, author = {Ronald R Coifman and S Lafon and A B Lee and Mauro Maggioni and B Nadler and Frederick Warner and Steven W Zucker}, year = {2005}, date = {2005-01-01}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, volume = {102}, number = {21}, pages = {7432–7438}, keywords = {diffusion geometry, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

Mahadevan, Sridhar; Maggioni, Mauro Value Function Approximation with Diffusion Wavelets and Laplacian Eigenfunctions Inproceedings University of Massachusetts, Department of Computer Science Technical Report TR-2005-38; Proc. NIPS 2005, 2005. BibTeX | Tags: diffusion geometry, diffusion wavelets, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory @inproceedings{smmm:ValueFunction, title = {Value Function Approximation with Diffusion Wavelets and Laplacian Eigenfunctions}, author = {Sridhar Mahadevan and Mauro Maggioni}, year = {2005}, date = {2005-01-01}, booktitle = {University of Massachusetts, Department of Computer Science Technical Report TR-2005-38; Proc. NIPS 2005}, keywords = {diffusion geometry, diffusion wavelets, Laplacian eigenfunctions, Machine learning, Manifold Learning, random walks, reinforcement learning, representation learning, spectral graph theory}, pubstate = {published}, tppubtype = {inproceedings} } |

Maggioni, Mauro; Bremer, James Jr. C; Coifman, Ronald R; Szlam, Arthur D Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs Conference 5914 (1), SPIE, San Diego, CA, USA, 2005. Links | BibTeX | Tags: diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory @conference{MBCS:BiorthogonalDiffusionWavelets, title = {Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs}, author = {Mauro Maggioni and James Jr. C Bremer and Ronald R Coifman and Arthur D Szlam}, editor = {Manos Papadakis and Andrew F Laine and Michael A Unser}, url = {http://link.aip.org/link/?PSI/5914/59141M/1}, year = {2005}, date = {2005-01-01}, journal = {Wavelets XI}, volume = {5914}, number = {1}, pages = {59141M}, publisher = {SPIE}, address = {San Diego, CA, USA}, keywords = {diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory}, pubstate = {published}, tppubtype = {conference} } |

Szlam, Arthur D; Maggioni, Mauro; Coifman, Ronald R; Bremer, James Jr. C Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions Conference 5914-1 , SPIE, San Diego, CA, USA, 2005. Links | BibTeX | Tags: diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory @conference{MSCB:MultiscaleManifoldMethods, title = {Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions}, author = {Arthur D Szlam and Mauro Maggioni and Ronald R Coifman and James Jr. C Bremer}, editor = {Manos Papadakis and Andrew F Laine and Michael A Unser}, url = {http://link.aip.org/link/?PSI/5914/59141D/1}, year = {2005}, date = {2005-01-01}, journal = {Wavelets XI}, volume = {5914-1}, pages = {59141D}, publisher = {SPIE}, address = {San Diego, CA, USA}, keywords = {diffusion geometry, diffusion wavelets, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory}, pubstate = {published}, tppubtype = {conference} } |

## 2004 |

Coifman, Ronald R; Maggioni, Mauro Multiresolution Analysis associated to diffusion semigroups: construction and fast algorithms Technical Report Dept. Comp. Sci., Yale University (YALE/DCS/TR-1289), 2004. BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems @techreport{CMTech, title = {Multiresolution Analysis associated to diffusion semigroups: construction and fast algorithms}, author = {Ronald R Coifman and Mauro Maggioni}, year = {2004}, date = {2004-05-01}, number = {YALE/DCS/TR-1289}, institution = {Dept. Comp. Sci., Yale University}, keywords = {diffusion geometry, Machine learning, Manifold Learning, multiscale analysis, random walks, spectral graph theory, stochastic systems}, pubstate = {published}, tppubtype = {techreport} } |

- Lectures at Summer School at Peking University, July 2017.
- PCMI Lectures, Summer 2016: Lecture 1, Lecture 2, Problems/discussion points
- Google Scholar
- Papers on the ArXiv
- Papers on MathsciNet
- Tutorials on diffusion geometry and multiscale analysis on graphs at the MRA Internet Program at IPAM: Part I and Part II.
- Diffusion Geometries, Diffusion Wavelets and Harmonic Analysis of large data sets, IPAM, Multiscale Geometric Analysis Program, Fall 2004.
- Diffusion Geometries, global and multiscale, IPAM, 2005.