Jared Tanner, Ph.D - US grants

Postdoctoral Research Fellowship

This grant supplies funds to cover shared local accommodations and meals for approximately fifteen US based junior researchers to attend the 2007 AMS von Neumann Symposium "Sparse Representation and High-Dimensional Geometry" being held from July 8th through the 12th, 2007. Junior researcher is defined as a graduate student, or researcher who has received their PhD. during or after 2002.
This conference is the first major conference in the USA dedicated to the emerging topic of sparse approximation and as such is expected to play an important role in the development of a cohesive research community. With its relationship to fundamental questions in pure mathematics as well as important implications for information acquisition, sparse approximation has received a great deal of interest from both mathematicians and engineers. The conference will be focused around the themes of: fundamental frameworks for where sparsity is present and can be exploited, application areas and practical considerations, the underlying high-dimensional geometric phenomenon allowing for sparse approximation algorithms to work, the design and analysis of computationally efficient algorithms, and presentations of application results. The conference is truly interdisciplinary, including participants from google and other industries, theoretical mathematicians studying banach spaces, and leading electrical engineers.

This grant supplies funds to allow approximately fifteen US based junior researchers to attend the first conference in the USA dedicated to sparse approximation; for details of this conference see "www.ams.org/meetings/vonneumann07.html." This topic concerns the development of mathematical tools to simplify large data sets, through the construction of good approximations which are as parsimonious as the data allows. These tools are increasingly important in the modern scientific environment characterized by ever larger data sets. Applications already under investigation include dynamic visualizations of the human beating heart, next generation circuit components for wireless communications, and faster sequencing of biological micro-arrays. Funds from this grant will allow for the rapid training of early stage researchers in this topic, increasing the rate and quality of expected advances in both its theory and application.

A proliferation of technical instruments, computational resources, and digital media are opening up new opportunities in security, entertainment, science and medicine. We are automatically acquiring large databases of measurements that span physical scales from the molecule, to the cell, tissue, organism, ecosystem, and beyond to stars and galaxies. However, the rate of acquisition of these data exceeds our ability to analyze them, and many applications are stymied not by the lack of data but by the daunting task of understanding it all. This project addresses the problem of constructing statistical models of such data sets using examples from the data itself. The researchers will study the technical problems associated with constructing statistical modeling in high-dimensional spaces from mathematical and computational points of view. This research will focus on applications of these methods to problems in multidimensional image analysis, including denoising, segmentation, and image synthesis.

This project will develop new algorithms for analyzing large sets of data. These algorithms make very weak assumptions about the statistical structure of data, and instead learn the statistics through large sets of examples, using raw measurements from very high dimensional spaces. The project combines a small team of researchers from probability theory, applied mathematics, and computer science to simultaneously address several important, fundamental questions pertaining to estimates of high-dimensional probability density functions, the computational challenges associated with these estimations, and the applications of these ideas to ongoing problems in science and medicine. The work in probability theory addresses basic questions about probability density functions in very high dimensional spaces. The applied mathematics work is developing methods for approximating these densities through lower-dimensional or sparse representations. The computer science research is examining implementation issues and the application of these methods to a wide range of ongoing problems that demand better algorithms for data analysis.