CAREER: Foundations for Geometric Analysis of Noisy Data

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An important role of computational geometry is to understand and formalize the structure of data. And as data is becoming a central currency of modern science, this role is growing in importance. However, much of classical computational geometry inherently assumes that all aspects of data are known and precise. This is rarely the case in practice. This project focuses on building the foundations for two extensions to classic geometric settings pertinent to noisy data. 1. The PI will study locational uncertainty in point sets, where the location of each data point is described by a probability distribution. Given such an input, the goal is to formalize how to construct, approximate, and concisely represent the distribution of geometric queries on this uncertain data. 2. The PI will study the geometric consequences of applying a statistical kernel (e.g. a Gaussian kernel) to a data set. He will investigate how this process can smooth data, remove degeneracies, and implicitly simplify and regularize algorithms. Moreover, he will explore the geometric structure of the resulting kernel density estimate, and how it relates to algorithms for the data and approximate representations of the data. The PI will lead the development of a data-focused educational program around the themes of data analysis, algorithmics, and visualization. The PI is developing a model course for this program on data mining; it focuses on the geometric, statistical, and algorithmic properties of data. An extensive set of course notes is being compiled, accompanied with videotaped lectures freely available online. This class and program attract many interdisciplinary and diverse students and observers. This program is part of a larger effort to make relevant data analysis techniques from computational geometry available to a broader data-rich audience.