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Mauro Maggioni, Professor of Mathematics and Computer Science and Electrical and Computer Engineering

Mauro Maggioni

Harmonic analysis, spectral graph theory, multiscale analysis, signal processing, applications to machine learning, Markov decision processes, imaging.

Contact Info:
Office Location:  117 Physics Building
Office Phone:  (919) 660-2825
Email Address: send me a message
Web Page:  http://www.math.duke.edu/~mauro

Teaching (Fall 2014):

  • MATH 465.01, INTRO HIGH DIM DATA ANALYSIS Synopsis
    SEE INSTRU, MW 03:05 PM-04:20 PM
  • MATH 561.01, SCIENTIFIC COMPUTING Synopsis
    SEE INSTRU, MW 01:25 PM-02:40 PM
  • MATH 790-90.04, MINICOURSE IN ADVANCED TOPICS Synopsis
    Gross Hall 304B, MWF 10:20 AM-11:10 AM
Education:

PhDWashington University, St. Louis2002
MSWashington University, St. Louis2000
Laurea in MatematicaUniversita' degli Studi di Milano, Italy1999
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 Processes

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.

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:

Harmonic • Multiscale • Spectral graph theory • Multiscale Dynamical systems • Laplacian • Hyperspectral • Imaging

Curriculum Vitae
Current Ph.D. Students   (Former Students)

  • Miles M Crosskey  
Postdocs Mentored

  • Wenjing Liao (August, 2013 - present)  
  • David Lawlor (2012/10-present)  
  • Joshua Vogelstein (2012/10-present)  
  • Samuel Gerber (2012/10-present)  
  • Grace Yi Wang (September, 2012 - present)  
  • Nate Strawn (2011 - present)  
  • 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   (More Publications)

  1. W.K. Allard, G. Chen, M. Maggioni, Multiscale Geometric Methods for Data Sets II: Geometric Wavelets, Appl. Comp. Harm. Anal., vol. 32 no. 3 (2012)
  2. M. Iwen, M. Maggioni, Approximation of Points on Low-Dimensional Manifolds Via Random Linear Projections, Inference & Information, vol. 2 (February, 2013) [doi]
  3. M. Maggioni, Geometric Estimation of Probability Measures in High-Dimensions, Proc. IEEE Asilomar Conference (November, 2013)
  4. S. Gerber, M. Maggioni, Multiscale dictionaries, transforms, and learning in high-dimensions, Proc. SPIE 8858, Wavelets and Sparsity XV (2013) [doi]
  5. Mauro Maggioni and Gustave L. Davis and Frederick J. Warner and Frank B. Geshwind and Andreas C. Coppi and Richard A. DeVerse and Ronald R. Coifman, Hyperspectral microscopic analysis of normal, benign and carcinoma microarray tissue sections, edited by Robert R. Alfano and Alvin Katz, Optical Biopsy VI, vol. 6091 no. 1 (2006), pp. 60910I, SPIE [1]
  6. Ronald R Coifman and Mauro Maggioni, Diffusion Wavelets, Appl. Comp. Harm. Anal., vol. 21 no. 1 (2006), pp. 53--94
  7. J. Bouvrie, M. Maggioni, Efficient Solution of Markov Decision Problems with Multiscale Representations, Proc. 50th Annual Allerton Conference on Communication, Control, and Computing (2012)
  8. Sridhar Mahadevan and Mauro Maggioni, Proto-value Functions: A Spectral Framework for Solving Markov Decision Processes, submitted (2006)
  9. J. Bouvrie, M. Maggioni, Geometric Multiscale Reduction for Autonomous and Controlled Nonlinear Systems, in Proc. IEEE Conference on Decision and Control (CDC) (2012) [pdf]
  10. Mauro Maggioni and Sridhar Mahadevan, Multiscale Diffusion Bases for Policy Iteration in Markov Decision Processes, submitted (Submitted, 2006)
  11. 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
  12. G. Chen, M. Maggioni, Multiscale Analysis of Plane Arrangements, in Proc. C.V.P.R. (2011)
  13. Mary A. Rohrdanz, Wenwei Zheng, Mauro Maggioni,Cecilia Clementi, Determination of reaction coordinates via locally scaled diffusion map, JCP, vol. 134 no. 12 (2011), pp. 124116
  14. G. Chen, M. Maggioni, Multiscale Geometric Dictionaries for point-cloud data, Proc. SampTA 2011 (2011)
  15. Wenwei Zheng,Mary A. Rohrdanz,Mauro Maggioni, Cecilia Clementi, Polymer reversal rate calculated via locally scaled diffusion map, JCP, vol. 134 no. 14 (2011), pp. 144109
  16. G. Chen, M. Maggioni, Multiscale Geometric Wavelets for the Analysis of Point Clouds, Proc. CISS (February, 2010)
  17. P.W. Jones, M. Maggioni, R. Schul, Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian, Ann. Acad. Scient. Fen., vol. 35 (January, 2010), pp. 1-44 [1975]  [abs]
  18. J. Lee, M. Maggioni, Multiscale Analysis of Time Series of Graphs, Proc. SampTA 2011 (2010)
  19. P.W. Jones, M. Maggioni, R. Schul, Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels, Proc. Nat. Acad. Sci., vol. 105 no. 6 (2008)
  20. R R Coifman, I G Kevrekidis, S Lafon, M Maggioni, B. Nadler, Diffusion Maps, reduction coordinates and low dimensional representation of stochastic systems, J.M.M.S., vol. 7 (2008), pp. 842-864
  21. S. Mahadevan, M. Maggioni, Proto-value Functions: A Laplacian Framework for Learning Representation and Control, Journ. Mach. Learn. Res. no. 8 (September, 2007)
  22. G. Chen, A.V. Little, M. Maggioni, Multi-Resolution Geometric Analysis for Data in High Dimensions, in Excursions in Harmonic Analysis, vol. 1 (2013), Birkha├╝ser Boston, ISBN 978-0-8176-8375-7 [doi]
  23. Sridhar Mahadevan and Mauro Maggioni, 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 (2005)
  24. Nets Hawk Katz and Elliot Krop and Mauro Maggioni, On the box problem, Math. Research Letters, vol. 4 (2002), pp. 515-519
  25. Ronald R Coifman and Stephane Lafon and Ann Lee and Mauro Maggioni and Boaz Nadler and Frederick Warner and Steven Zucker, Geometric Diffusions as a tool for Harmonic Analysis and structure definition of data. Part II: Multiscale methods, Proc. of Nat. Acad. Sci. no. 102 (2005), pp. 7432--7438
  26. Ronald R Coifman and Stephane Lafon and Ann Lee and Mauro Maggioni and Boaz Nadler and Frederick Warner and Steven Zucker, Geometric Diffusions as a tool for Harmonic Analysis and structure definition of data. Part I: Diffusion maps, Proc. of Nat. Acad. Sci. no. 102 (2005), pp. 7426--7431
  27. J. Bouvrie, M. Maggioni, Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning (Submitted, 2012) [1212.1143]
Recent Grant Support

  • Information, Approximation, and Fast Algorithms for Data in High Dimensions, Air Force Office of Scientific Research, 2013/12-2016/12.      
  • ATD: Online Multiscale Algorithms for Geometric Density Estimation in High-Dimensions and Persistent Homology of Data for Improved Threat, National Science Foundation, DMS-1222567, 2012/09-2016/08.      
  • EMSW21-RTG: Geometric, Topological and Statistical Methods for Analyzing Massive Datasets, National Science Foundation, DMS-1045153, 2011/08-2016/07.      
  • A Rigorous Statistical Framework for the Mathematics of Sensing, Exploitation and Execution, Defense Advanced Research Projects Agency, 2011/11-2014/11.      
  • X-Ray Scatter and Phase Imaging for Explosive Detection, US Department of Homeland Security, HSHQDC-11-C-00083, 2011/09-2014/09.      
  • CAREER: Multiscale methods for high-dimensional data, graphs and dynamical systems, National Science Foundation, DMS-0847388, 2009/07-2014/06.      
  • Knowledge Enhanced Exapixel Photography, Defense Advanced Research Projects Agency, N66001-11-1-4002, 2010/11-2013/11.      
  • Multiscale geometry for the analysis of high dimensional datasets, Washington State University, 113054 G002745, 2010/05-2013/05.      
  • NeTS: Small: Collaborative Research: Online Social Networks: Measurement and Characterization Methodologies, National Science Foundation, IIS-0916855, 2009/09-2012/08.      
  • Mathematical Foundations of Multiscale Graph Representations and Interactive Learning, National Science Foundation, CCF-0808847, 2008/05-2012/07.      
Conferences Organized

  • 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  
  • Large Data Workshop, C.T.M.S., Duke, November 13, 2009  
  • A.A.A.I. workshop on manifold learning, November 5, 2009  
  • Internet Multi-Resolution Analysis: Foundations, Applications and Practice, Organizer, September, 2008 - December, 2008  
  • Workshop on Eigenfunctions of the Laplacian, ICIAM 2007, July 18, 2007  
  • Document Space, Organizer, January, 2006  

 

dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320