# 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

## 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.

## Awards, Honors, and Distinctions

- AMS Fellow. AMS. 2013
- Sloan Fellowship. Sloan Foundation. 2008
- Sloan Research Fellowship-Mathematics. Alfred P. Sloan Foundation. 2008
- Popov Prize. Popov Foundation. 2007

## 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: Numerical Linear Algebra, Optimization and Monte Carlo Simulation
- MATH 690-40: Topics in Probability Theory
- MATH 790-90: Minicourse in Advanced Topics
- MATH 799: Special Readings

## Representative Publications

- Maggioni, M; Minsker, S; Strawn, N,
*Multiscale dictionary learning: Non-asymptotic bounds and robustness*, Journal of machine learning research : JMLR, vol 17 (2016) [abs]. - Maggioni, M,
*Geometry of Data and Biology*, Notices of the American Mathematical Society, vol 62 no. 10 (2015), pp. 1185-1188 [10.1090/noti1289] [abs]. - Maggioni, M; Minsker, S; Strawn, N,
*Geometric multi-resolution analysis for dictionary learning*, Proceedings of SPIE - The International Society for Optical Engineering, vol 9597 (2015) [10.1117/12.2189594] [abs]. - Altemose, N; Miga, KH; Maggioni, M; Willard, HF,
*Genomic characterization of large heterochromatic gaps in the human genome assembly.*, PLoS computational biology, vol 10 no. 5 (2014) [abs]. - Coppola, A; Wenner, BR; Ilkayeva, O; Stevens, RD; Maggioni, M; Slotkin, TA; Levin, ED; Newgard, CB,
*Branched-chain amino acids alter neurobehavioral function in rats.*, American journal of physiology. Endocrinology and metabolism, vol 304 no. 4 (2013), pp. E405-E413 [10.1152/ajpendo.00373.2012] [abs].