AF: Small: The Geometry of Learning on Structured Data Objects

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The project builds connections between computational geometry and machine learning by extending and demonstrating how advances in geometry can impact and inform machine learning, and vice versa. Computational geometry offers many ways to characterize and compute on shapes in low dimensions (such as spatially-encoded trajectories of people or objects) and provides understanding about point sets in high dimensions. Machine learning focuses on automatically discovering patterns which generalize across a population to new data. This project will leverage computational geometry insights to make analysis of low-dimensional shapes more amenable to machine learning, and to improve understanding and efficacy of high-dimensional data analysis. The project will also help bridge these connections by supporting several events including international workshops on geometry in machine learning, and community building ones in data science. In more detail, this project highlights two keystone applications. The first is dealing with trajectories, which can be modeled as time-parameterized curves in a spatial domain and pose many challenges for direct use within typical analysis pipelines. The developed approach shows how to convert such objects (and related low-dimensional geometric structures) into a vectorized representation so it can seamlessly integrate with numerous data-analysis tasks. The second is in analyzing data sets already embedded as points in a high-dimensional space. Word vector representations are a central and important such example where the individual data-point coordinates do not have meaning. Hence, the project uses the geometry of these high-dimensional data sets to provide richer analytical operations and more intuitive and transparent analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.