## 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. |

## 2020 |

Fei Lu Mauro Maggioni, Sui Tang Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories Journal Article Forthcoming Foundation of Computational Mathematics, Forthcoming. Abstract | Links | BibTeX | Tags: agent-based models, interacting particle systems, Machine learning, statistics, stochastic systems @article{learningStochasticInteracting, title = {Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories}, author = {Fei Lu, Mauro Maggioni, Sui Tang}, url = {https://arxiv.org/abs/2007.15174}, year = {2020}, date = {2020-07-15}, journal = {Foundation of Computational Mathematics}, abstract = {We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of 1-dimensional non-parametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order 1/2 in terms of the time spacings between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel al- gorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model.}, keywords = {agent-based models, interacting particle systems, Machine learning, statistics, stochastic systems}, pubstate = {forthcoming}, tppubtype = {article} } We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of 1-dimensional non-parametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order 1/2 in terms of the time spacings between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel al- gorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model. |

## 2017 |

Crosskey, Miles C; Maggioni, Mauro ATLAS: A geometric approach to learning high-dimensional stochastic systems near manifolds Journal Article Journal of Multiscale Modeling and Simulation, 15 (1), pp. 110–156, 2017, (arxiv: 1404.0667). Links | BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, statistics, stochastic systems @article{CM:ATLAS, title = {ATLAS: A geometric approach to learning high-dimensional stochastic systems near manifolds}, author = {Miles C Crosskey and Mauro Maggioni}, url = {https://arxiv.org/abs/1404.0667 https://doi.org/10.1137/140970951}, year = {2017}, date = {2017-01-01}, journal = {Journal of Multiscale Modeling and Simulation}, volume = {15}, number = {1}, pages = {110–156}, note = {arxiv: 1404.0667}, keywords = {diffusion geometry, Machine learning, Manifold Learning, statistics, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

## 2013 |

Crosskey, Miles C; Maggioni, Mauro Learning of intrinsically low-dimensional stochastic systems in high-dimensions, I Technical Report 2013, (in preparation). BibTeX | Tags: Manifold Learning, stochastic systems @techreport{CM:MultiscaleDynamicsI, title = {Learning of intrinsically low-dimensional stochastic systems in high-dimensions, I}, author = {Miles C Crosskey and Mauro Maggioni}, year = {2013}, date = {2013-01-01}, note = {in preparation}, keywords = {Manifold Learning, stochastic systems}, pubstate = {published}, tppubtype = {techreport} } |

## 2011 |

Zheng, W; Rohrdanz, M A; Maggioni, Mauro; Clementi, Cecilia Polymer reversal rate calculated via locally scaled diffusion map Journal Article J. Chem. Phys., (134), pp. 144108, 2011. BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, molecular dynamics, stochastic systems @article{ZRMC:PolymerReversal, title = {Polymer reversal rate calculated via locally scaled diffusion map}, author = {W Zheng and M A Rohrdanz and Mauro Maggioni and Cecilia Clementi}, year = {2011}, date = {2011-01-01}, journal = {J. Chem. Phys.}, number = {134}, pages = {144108}, keywords = {diffusion geometry, Machine learning, Manifold Learning, molecular dynamics, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

Rohrdanz, M A; Zheng, W; Maggioni, Mauro; Clementi, Cecilia Determination of reaction coordinates via locally scaled diffusion map Journal Article J. Chem. Phys., (134), pp. 124116, 2011. BibTeX | Tags: diffusion geometry, Machine learning, Manifold Learning, molecular dynamics, stochastic systems @article{RZMC:ReactionCoordinatesLocalScaling, title = {Determination of reaction coordinates via locally scaled diffusion map}, author = {M A Rohrdanz and W Zheng and Mauro Maggioni and Cecilia Clementi}, year = {2011}, date = {2011-01-01}, journal = {J. Chem. Phys.}, number = {134}, pages = {124116}, keywords = {diffusion geometry, Machine learning, Manifold Learning, molecular dynamics, stochastic systems}, pubstate = {published}, tppubtype = {article} } |

## 2008 |

Coifman, Ronald R; Kevrekidis, Ioannis G; Lafon, Stephane; Maggioni, Mauro; Nadler, Boaz Diffusion Maps, reduction coordinates and low dimensional representation of stochastic systems Journal Article SIAM J.M.M.S., 7 (2), pp. 842–864, 2008. BibTeX | Tags: diffusion geometry, dynamical systems, Laplacian eigenfunctions, Machine learning, model reduction, stochastic systems @article{CKLMN:DiffusionMapsReductionCoordinates, title = {Diffusion Maps, reduction coordinates and low dimensional representation of stochastic systems}, author = {Ronald R Coifman and Ioannis G Kevrekidis and Stephane Lafon and Mauro Maggioni and Boaz Nadler}, year = {2008}, date = {2008-01-01}, journal = {SIAM J.M.M.S.}, volume = {7}, number = {2}, pages = {842–864}, keywords = {diffusion geometry, dynamical systems, Laplacian eigenfunctions, Machine learning, model reduction, stochastic systems}, 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} } |

## 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} } |

## 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} } |

## 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.