CAREER: Learning from Coarse, Nonmetric, and Incomplete Data

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In recent years, we have witnessed an explosion in the amounts of data being acquired and analyzed in a wide variety of contexts. Data-driven techniques are increasingly applied, not only in the traditional quantitative sciences, but also throughout the social sciences and in a variety of other non-traditional scenarios that challenge many of our common assumptions. For example, in contexts such as collaborative filtering, personalized and predictive medicine, and personalized learning systems, we face a variety of challenges due largely to the fact that an important -- often the only -- source of data is people. In these and many other modern applications, we want to learn about people using the data that people supply.This presents several difficulties, including the fact that such data is often very "coarse" or heavily "quantized". It might even be binary or entirely nonmetric data consisting of categories or comparisons. Moreover, in many of these cases it is impossible to fully sample the data, and the underlying data of interest may be constantly changing, necessitating approaches that can handle incomplete observations and dynamic data models. This research confronts these difficulties by building on recent progress in the design of efficient algorithms for exploiting low-dimensional structure to perform inference, often using highly incomplete and coarse observations. This research addresses a number of fundamental theoretical and algorithmic questions in the context of low-rank matrix recovery, nonmetric multidimensional scaling, unfolding, and low-dimensional dynamic models. It has applications in contexts such as collaborative filtering, personalized and predictive medicine, and personalized learning systems.