Mauro Maggioni, Assistant Professor of Mathematics and Computer Science

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

Contact Info:

Office Location:	293 Physics
Office Phone:	(919)-660-2825
Email Address:
Web Page:	http://www.math.duke.edu/~mauro

Education:

PhD	Washington University, St. Louis	2002
MS	Washington University, St. Louis	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 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)

• Anna V. Little  

Representative Publications   (More Publications)

1. Ronald R Coifman and Mauro Maggioni, Diffusion Wavelets, Appl. Comp. Harm. Anal., vol. 21 no. 1 (2006), pp. 53--94
2. 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
3. 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
4. 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]
5. Nets Hawk Katz and Elliot Krop and Mauro Maggioni, On the box problem, Math. Research Letters, vol. 4 (2002), pp. 515-519
6. Arthur D. Szlam and Mauro Maggioni and Ronald R. Coifman, A General Framework for Adaptive Regularization Based on Diffusion Processes On Graphs no. YALE/DCS/TR1365 (Submitted, 2006)
7. Sridhar Mahadevan and Mauro Maggioni, Proto-value Functions: A Spectral Framework for Solving Markov Decision Processes, submitted (2006)
8. Mauro Maggioni and Sridhar Mahadevan, Multiscale Diffusion Bases for Policy Iteration in Markov Decision Processes, submitted (Submitted, 2006)
9. Michael W Mahoney and Mauro Maggioni and Petros Drineas,, Tensor-CUR Decompositions For Tensor-Based Data, in Proc 12-th Annual SIGKDD, to appear in SIAM Journal on Matrix Analysis and Applications (Accepted, 2006)

Recent Grant Support

• ONR BAA 07-001, ONR, N000140710625, 2007/04-2010/06.      
• Geometric Modeling OF and Function Approximation ON High Dimensional, Multi-Modal Data Sets, Office of Naval Research, N00014-07-1-0625, 2007/02-2010/03.      
• Diffusion Multiscale Analysis, National Science Foundation, DMS-0650413, 2006/08-2008/06.      
• Diffusion Multiscale Analysis, NSF DMS, 0650413, 2006/08-2008/06.      
• Subcontract for L.A.C.O.S.T.E. Program, 2006/08-2007/12.      
• Diffusion Multiscale Analysis, National Science Foundation, DMS 0501250, 2005/07-2006/07.      

Conferences Organized

• Workshop on Eigenfunctions of the Laplacian, ICIAM 2007, July 18, 2007  
• Document Space, Organizer, January, 2006