I am interested in novel constructions inspired by classical harmonic analysis that allow to analyse the geometry of manifolds and graphs and functions on such structures. These constructions are motivated by several important applications across many fields. In many situations we are confronted with large amounts of apparently unstructured high-dimensional data. I find fascinating to study the intrinsic geometry of such data, and exploiting in order to study, explore, visualize, characterize statistical properties of the data. Oftentimes such data is modeled as a manifold (or something "close to a manifold") or a graph, and functions on these spaces need to approximated or "learned" from the data and experiments on the data. For example each data point could be a document, a graph associated with the documents could be given by for example hyperlinks, or by similarity of word frequencies, and a function on the set of documents would be how interesting I personally score a document. One may wish to learn how to predict how much I would score documents I have not seen yet. This can be cast as an approximation problem on the graph of documents, and it turns out that one can generalize Euclidean-type approximation techniques (in particular multiscale regression techniques) to tackle this problem. An application of the above techniques that I find particularly interesting is Markov Decision Processes and Reinforcement Learning, where the problem of learning a behaviour from experience is cast in a rather general optimization and learning framework that involves approximations of functions and operators on graphs and manifolds. I am also interested in imaging, in particular I am working on novel classes of nonlinear denoising algorithms, based on diffusion processes on graphs of features built from images. Another interest is in the geometry of multiscale dynamical systems, and the construction of algorithms for the empirical construction of approximate equations for such systems. I also work on hyperspectral imaging, in particular in building automatic classifiers for discriminating normal from cancerous biopsies, for automated diagnostics and pathology.

Office Location: | 117 Physics Bldg, Office #293, Durham, NC 27708 |

Office Phone: | (919) 660-2825 |

Email Address: | |

Web Page: | http://www.math.duke.edu/~mauro |

**Education:**Ph.D. Washington University 2002 M.S. Washington University 2000 Laurea in Matematica Universita' degli Studi di Milano, Italy 1999

**Specialties:**-
Applied Math

Analysis

Probability

**Research Interests:***Harmonic analysis, with applications to statistical analysis of high-dimensional data, machine learning, imaging.***Current projects:**Multiscale analysis on graphs and manifolds, Nonlinear image denoising, Compressed imaging and hyperspectral imaging, Supervised and semisupervised learning on manifolds and graphs, Universal mappings via the eigenfunctions of the Laplacian, Perturbation of eigenfunctions of the Laplacian on graphs, Multiscale manifolds methods for Markov Decision ProcessesI am interested in novel constructions inspired by classical harmonic analysis that allow to analyse the geometry of manifolds and graphs and functions on such structures. These constructions are motivated by several important applications across many fields. In many situations we are confronted with large amounts of apparently unstructured high-dimensional data. I find fascinating to study the intrinsic geometry of such data, and exploiting in order to study, explore, visualize, characterize statistical properties of the data. Oftentimes such data is modeled as a manifold (or something "close to a manifold") or a graph, and functions on these spaces need to approximated or "learned" from the data and experiments on the data. For example each data point could be a document, a graph associated with the documents could be given by for example hyperlinks, or by similarity of word frequencies, and a function on the set of documents would be how interesting I personally score a document. One may wish to learn how to predict how much I would score documents I have not seen yet. This can be cast as an approximation problem on the graph of documents, and it turns out that one can generalize Euclidean-type approximation techniques (in particular multiscale regression techniques) to tackle this problem. An application of the above techniques that I find particularly interesting is Markov Decision Processes and Reinforcement Learning, where the problem of learning a behaviour from experience is cast in a rather general optimization and learning framework that involves approximations of functions and operators on graphs and manifolds. I am also interested in imaging, in particular I am working on novel classes of nonlinear denoising algorithms, based on diffusion processes on graphs of features built from images. Another interest is in the geometry of multiscale dynamical systems, and the construction of algorithms for the empirical construction of approximate equations for such systems. I also work on hyperspectral imaging, in particular in building automatic classifiers for discriminating normal from cancerous biopsies, for automated diagnostics and pathology.

**Areas of Interest:**Harmonic analysis

Multiscale analysis

Markov decision processes

Machine learning

High-dimensional data analysis

Stochastic dynamical systems

Signal processing

Imaging (e.g. hyperspectral)

Geometric measure theory

**Keywords:**Algorithms • Amino Acids, Branched-Chain • Animals • Anxiety • Base Sequence • Behavior, Animal • Brain • Chemistry, Physical • Chromosome Mapping • Diet • Diet, High-Fat • Dietary Sucrose • Diffusion • Exploratory Behavior • Face • Harmonic • Hyperspectral • Imaging • Imaging, Three-Dimensional • Kinetics • Kynurenic Acid • Laplacian • Male • Models, Neurological • Models, Theoretical • Molecular Conformation • Molecular Dynamics Simulation • Mood Disorders • Multiscale • Multiscale Dynamical systems • Nanopores • Neural Networks (Computer) • Neurons • Obesity • Polymers • Rats • Rats, Wistar • Serotonin • Spectral graph theory • Thermodynamics • Time Factors • Tryptophan • Weight Gain

**Current Ph.D. Students**- Shan Shan
- Miles M Crosskey

**Postdocs Mentored**- James Murphy (2015/12-present)
- Stefano Vigogna (January, 2015 - present)
- Wenjing Liao (August, 2013 - present)
- David Lawlor (2012 - 2015)
- Joshua Vogelstein (2012 - 2014)
- Samuel Gerber (2012 - 2015)
- Grace Yi Wang (September, 2012 - present)
- Nate Strawn (2011 - 2014)
- Mark Iwen (2010 - 2013)
- Guangliang Chen (2009 - 2012)
- Jake Bouvrie (2009 - 2012)
- Yoon-Mo Jung (2007 - 2009)

**Undergraduate Research Supervised**- Jason Lee (May, 2009 - May, 2010)

**Representative Publications**- Allard, WK; Chen, G; Maggioni, M,
*Multiscale Geometric Methods for Data Sets II: Geometric Wavelets*, CoRR, vol. abs/1105.4924 no. 3 (2012) - Iwen, MA; Maggioni, M,
*Approximation of Points on Low-Dimensional Manifolds Via Random Linear Projections*, vol. 2 (February, 2013) [1204.3337v1], [doi] [abs] - M. Maggioni,
*Geometric Estimation of Probability Measures in High-Dimensions*, Proc. IEEE Asilomar Conference (November, 2013) - Gerber, S; Maggioni, M,
*Multiscale dictionaries, transforms, and learning in high-dimensions*, Smart Structures and Materials 2005: Active Materials: Behavior and Mechanics, vol. 8858 (2013), ISSN 0277-786X [Gateway.cgi], [doi] - Maggioni, M; Davis, GL; Warner, FJ; Geshwind, FB; Coppi, AC; DeVerse, RA; Coifman, RR,
*Hyperspectral microscopic analysis of normal, benign and carcinoma microarray tissue sections*, edited by Robert R. Alfano and Alvin Katz, Progress in Biomedical Optics and Imaging Proceedings of Spie, vol. 6091 no. 1 (2006), pp. 60910I, SPIE, ISSN 1605-7422 [1], [doi] [abs] - Coifman, RR; Maggioni, M,
*Diffusion wavelets*, Applied and Computational Harmonic Analysis, vol. 21 no. 1 (July, 2006), pp. 53-94, ISSN 1063-5203 [doi] [abs] - Bouvrie, J; Maggioni, M,
*Efficient solution of Markov decision problems with multiscale representations*, 2012 50th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2012 (December, 2012), pp. 474-481 [doi] [abs] - Sridhar Mahadevan and Mauro Maggioni,
*Proto-value Functions: A Spectral Framework for Solving Markov Decision Processes*, submitted (2006) - Bouvrie, JV; Maggioni, M,
*Geometric multiscale reduction for autonomous and controlled nonlinear systems.*, in Proc. IEEE Conference on Decision and Control (CDC), CDC (2012), pp. 4320-4327, IEEE, ISBN 978-1-4673-2065-8 [mostRecentIssue.jsp], [doi] - Mauro Maggioni and Sridhar Mahadevan,
*Multiscale Diffusion Bases for Policy Iteration in Markov Decision Processes*, submitted (Submitted, 2006) - G. Chen, A.V. Little, M. Maggioni, L. Rosasco,
*Some recent advances in multiscale geometric analysis of point clouds*, in Wavelets and Multiscale Analysis: Theory and Applications (March, 2011), Springer - G. Chen, M. Maggioni,
*Multiscale Analysis of Plane Arrangements*, in Proc. C.V.P.R. (2011) - Rohrdanz, MA; Zheng, W; Maggioni, M; Clementi, C,
*Determination of reaction coordinates via locally scaled diffusion map.*, The Journal of Chemical Physics, vol. 134 no. 12 (2011), pp. 124116 [21456654], [doi] [abs] - G. Chen, M. Maggioni,
*Multiscale Geometric Dictionaries for point-cloud data*, Proc. SampTA 2011 (2011) - Zheng, W; Rohrdanz, MA; Maggioni, M; Clementi, C,
*Polymer reversal rate calculated via locally scaled diffusion map.*, The Journal of Chemical Physics, vol. 134 no. 14 (2011), pp. 144109 [21495744], [doi] [abs] - Chen, G; Maggioni, M,
*Multiscale geometric wavelets for the analysis of point clouds*, 2010 44th Annual Conference on Information Sciences and Systems, CISS 2010 (February, 2010) [doi] [abs] - Jones, PW; Maggioni, M; Schul, R,
*Universal local parametrizations via heat kernels and eigenfunctions of the laplacian*, Annales Academiae Scientiarum Fennicae Mathematica, vol. 35 no. 1 (January, 2010), pp. 131-174, ISSN 1239-629X [0709.1975v4], [doi] [abs] - J. Lee, M. Maggioni,
*Multiscale Analysis of Time Series of Graphs*, Proc. SampTA 2011 (2010) - Jones, PW; Maggioni, M; Schul, R,
*Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels.*, Proceedings of the National Academy of Sciences of the United States of America, vol. 105 no. 6 (2008), pp. 1803-1808 [18258744], [doi] [abs] - Coifman, RR; Kevrekidis, IG; Lafon, S; Maggioni, M; Nadler, B,
*Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems*, Multiscale Modeling & Simulation, vol. 7 no. 2 (2008), pp. 842-864, ISSN 1540-3459 [doi] [abs] - Mahadevan, S; Maggioni, M,
*Proto-value Functions: A Laplacian Framework for Learning Representation and Control*, Journ. Mach. Learn. Res. no. 8 (September, 2007) - Chen, G; Little, AV; Maggioni, M,
*Multi-resolution geometric analysis for data in high dimensions*, in Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center, vol. 1 (January, 2013), pp. 259-285, Birkhaüser Boston, ISBN 9780817683764 [doi] [abs] - Mahadevan, S; Maggioni, M,
*Value function approximation with diffusion wavelets and Laplacian eigenfunctions*, in University of Massachusetts, Department of Computer Science Technical Report TR-2005-38; Proc. NIPS 2005, Advances in Neural Information Processing Systems (December, 2005), pp. 843-850, ISSN 1049-5258 [abs] - Katz, NH; Krop, E; Maggioni, M,
*On the box problem*, Math. Research Letters, vol. 4 (2002), pp. 515-519 - Coifman, RR; Lafon, S; Lee, AB; Maggioni, M; Nadler, B; Warner, F; Zucker, SW,
*Geometric diffusions as a tool for harmonic analysis and structure definition of data: multiscale methods.*, Proceedings of the National Academy of Sciences of the United States of America, vol. 102 no. 21 (May, 2005), pp. 7432-7437, ISSN 0027-8424 [15899969], [doi] [abs] - Coifman, RR; Lafon, S; Lee, AB; Maggioni, M; Nadler, B; Warner, F; Zucker, SW,
*Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps.*, Proceedings of the National Academy of Sciences of the United States of America, vol. 102 no. 21 (May, 2005), pp. 7426-7431, ISSN 0027-8424 [15899970], [doi] [abs] - J. Bouvrie, M. Maggioni,
*Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning*(Submitted, 2012) [1212.1143]

- Allard, WK; Chen, G; Maggioni, M,

**Recent Grant Support***Dimension Reduction for Open Quantum Systems*, Stanford University, 2016/04-2020/03.*BIGDATA:Collaborative Research:F:From Data Geometries to Information Networks*, National Science Foundation, 1546392, 2016/01-2019/12.*Statistical learning for high-dimensional stochastic dynamical systems*, National Science Foundation, 1522651, 2015/09-2018/08.*EMSW21-RTG: Geometric, Topological*, National Science Foundation, 2011/08-2018/07.*Collaborative Research: SI2-CHE-ExTASY Extensible Tools for Advanced Sampling and AnalYsis*, National Science Foundation, CHE-1265920, 2013/09-2017/08.*ATD: Online Multiscale Algorithms for Geometric Density Estimation in High-Dimensions and Persistent Homology of Data fo*, National Science Foundation, DMS-1222567, 2012/09-2017/08.*ATD: Online Multiscale Algorithms for Geometric Density Estimation in High-Dimensions and Persistent Homology of Data fo*, National Science Foundation, DMS-1222567, 2012/09-2017/08.*Structured Dictionary Models and Learning for High Resolution Images*, National Science Foundation, DMS-1320655, 2013/08-2017/07.*Structured Dictionary Models and Learning for High Resolution Images*, National Science Foundation, DMS-1320655, 2013/08-2017/07.*Information, Approximation, and Fast Algorithms for Data in High Dimensions*, Air Force Office of Scientific Research, 2013/12-2016/12.*Geometric Approximation and Estimation of Probability Measures in High Dimensions*, Air Force Office of Scientific Research, FA9550-14-1-0033, 2013/12-2016/12.*Multiscale Analysis of Dynamic Graphs*, Office of Naval Research, N00014-12-1-0601-01, 2012/08-2016/07.*Multiscale Analysis of Dynamic Graphs*, Office of Naval Research, N00014-12-1-0601-01, 2012/08-2016/07.

**Conferences Organized**- SAMSI-FODAVA Workshop on Interactive Visualization and Analysis of Massive Data. December 10, 2012, SAMSI-FODAVA Workshop on Interactive Visualization and Analysis of Massive Data, December 10, 2012
- Symposium of Knowledge Extraction at A.M.S. nat. meeing 2010. January 13, 2010, Symposium of Knowledge Extraction at A.M.S. nat. meeing 2010, January 13, 2010
- Large Data Workshop, C.T.M.S., Duke. November 13, 2009, Large Data Workshop, C.T.M.S., Duke, November 13, 2009
- A.A.A.I. workshop on manifold learning. November 5, 2009, A.A.A.I. workshop on manifold learning, November 5, 2009
- Organizer : Internet Multi-Resolution Analysis: Foundations, Applications and Practice. September 2008 - December 2008, Organizer : Internet Multi-Resolution Analysis: Foundations, Applications and Practice, September, 2008 - December, 2008
- Workshop on Eigenfunctions of the Laplacian, ICIAM 2007, July 18, 2007
- Document Space, Organizer, January, 2006