# Mauro Maggioni

### Professor in the Department of Mathematics

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.

###### Appointments and Affiliations

- Professor in the Department of Mathematics
- Professor in the Department of Electrical and Computer Engineering
- Professor in the Department of Computer Science

###### Contact Information:

**Office Phone:**(919) 660-2825**Web Page:**

###### Education:

- Ph.D. Washington University, 2002
- M.S. Washington University, 2000

###### Curriculum Vitae

###### Research Interests:

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.

###### Specialties:

Applied Math

Analysis

Probability

###### Courses Taught:

- COMPSCI 393: Research Independent Study
- MATH 431: Advanced Calculus I
- MATH 465: Introduction to High Dimensional Data Analysis
- MATH 493: Research Independent Study
- MATH 561: Scientific Computing
- MATH 690-40: Topics in Probability Theory
- MATH 790-90: Minicourse in Advanced Topics
- MATH 799: Special Readings

###### Representative Publications: (More Publications)

- 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). - M. Iwen, M. Maggioni,
*Approximation of Points on Low-Dimensional Manifolds Via Random Linear Projections*, Inference & Information, vol 2 (February, 2013) [doi]. - M. Maggioni,
*Geometric Estimation of Probability Measures in High-Dimensions*, Proc. IEEE Asilomar Conference (November, 2013). - S. Gerber, M. Maggioni,
*Multiscale dictionaries, transforms, and learning in high-dimensions*, Proc. SPIE 8858, Wavelets and Sparsity XV (2013) [doi]. - 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*, Optical Biopsy VI, vol 6091 no. 1 (2006), pp. 60910I [1]. - Ronald R Coifman and Mauro Maggioni,
*Diffusion Wavelets*, Appl. Comp. Harm. Anal., vol 21 no. 1 (2006), pp. 53--94. - J. Bouvrie, M. Maggioni,
*Efficient Solution of Markov Decision Problems with Multiscale Representations*, Proc. 50th Annual Allerton Conference on Communication, Control, and Computing (2012). - Sridhar Mahadevan and Mauro Maggioni,
*Proto-value Functions: A Spectral Framework for Solving Markov Decision Processes*, submitted (2006). - J. Bouvrie, M. Maggioni,
*Geometric Multiscale Reduction for Autonomous and Controlled Nonlinear Systems*(2012) [pdf]. - 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*(March, 2011). - G. Chen, M. Maggioni,
*Multiscale Analysis of Plane Arrangements*(2011). - 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. - G. Chen, M. Maggioni,
*Multiscale Geometric Dictionaries for point-cloud data*, Proc. SampTA 2011 (2011). - 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. - G. Chen, M. Maggioni,
*Multiscale Geometric Wavelets for the Analysis of Point Clouds*, Proc. CISS (February, 2010). - 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]. - J. Lee, M. Maggioni,
*Multiscale Analysis of Time Series of Graphs*, Proc. SampTA 2011 (2010). - 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). - 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. - S. Mahadevan, M. Maggioni,
*Proto-value Functions: A Laplacian Framework for Learning Representation and Control*, Journ. Mach. Learn. Res. no. 8 (September, 2007). - G. Chen, A.V. Little, M. Maggioni,
*Multi-Resolution Geometric Analysis for Data in High Dimensions*, vol 1 (2013) [doi]. - Sridhar Mahadevan and Mauro Maggioni,
*Value Function Approximation with Diffusion Wavelets and Laplacian Eigenfunctions*(2005). - Nets Hawk Katz and Elliot Krop and Mauro Maggioni,
*On the box problem*, Math. Research Letters, vol 4 (2002), pp. 515-519. - 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. - 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. - J. Bouvrie, M. Maggioni,
*Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning*(Submitted, 2012) [1212.1143].