RTG: Data-Oriented Mathematical and Statistical Sciences

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The need and desire to analyze copious volumes of disparate data result in significant challenges. This RTG (Research Training Group) program will create a research environment and associated curricular elements that will engage U.S. citizen and permanent resident trainees in activities that will foster an understanding of the roles that statistics, computational mathematics, and applied harmonic analysis play in addressing data-oriented problems and appreciation of the synergies that can manifest when ideas from these areas, which are often studied by separate groups of students with little crossover, are brought to bear simultaneously. Durable impact is sought at ASU through cultivation of crosscutting faculty collaborations and curricular innovations intended to stimulate long-term strength in rigorous, integrated data-oriented mathematical and statistical research. Aggressive dissemination of innovative elements of the program will seek to provide a model for modern integrated data-oriented mathematics training for other institutions, and we will launch the careers of young researchers who will carry this vision.The research will focus on the following three problem areas. (i) Closed-loop design of experiment for efficient data acquisition: Traditional approaches to collection and analysis of data are essentially "feed-forward" in nature, for example, data are collected, numerical algorithms are used to process it (e.g., for compression or to transform it in some other way), and statistical methods are ultimately employed for inference. Statisticians have long recognized the appeal of sequential design of experiments in which the nature of a sample is not fixed in advance, rather depends on previously observed samples. Recent and ongoing technological advances have led to measurement devices possessing many degrees of freedom that enable manipulating the nature of the measurement, often electronically and in real time. In this context, sequential design of experiments takes on a fundamental importance in throttling back otherwise unmanageable data torrents. Instead of collecting all the data all the time, a feedback strategy can be used to acquire only the most important new data for the task at hand based on what has already been observed; (ii) Data driven non-classical numerical approximation tools: A central problem in sensing and model simulation is recovery of characteristic features of a function, signal, image, or operator from a set (frequently these days a very large set) of collected data. In almost all such situations, the data set constitutes an incomplete and noisy description of the system. Classical numerical methods mostly use a limited system model as well data interpolation or approximation to extract pertinent model information and features. Deducing crucial features in large volumes of data calls out for new methods that are readily adaptable to model improvements and inclusion of appropriate prior or statistical information on provided data; (iii) Approximation for statistical inference: Traditional methods in approximation theory and their numerical realizations seek to reconstruct functions from measurements with fidelity described by a metric on a function space. Our RTG has particular expertise with problems where the data are not direct samples of the underlying function (e.g., they may be measurements of some transformed version of the function) and also where the objective is to reconstruct specific features of the function rather than the function itself. Earlier collaborations among the RTG PIs has led to the use of both overcomplete representations (e.g., frames) and statistical ideas in this vein of research.