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

该项目通过扩展和展示几何的进步如何影响和告知机器学习，以及反之亦然，在计算几何和机器学习之间建立了联系。计算几何提供了许多方法来表征和计算低维形状(如人或物体的空间编码轨迹)，并提供了对高维点集的理解。机器学习的重点是自动发现模式，这些模式在整个群体中概括为新数据。这个项目将利用计算几何的洞察力，使低维形状的分析更易于机器学习，并提高对高维数据分析的理解和效率。该项目还将通过支持几项活动，包括机器学习中的几何国际研讨会和数据科学中的社区建设，来帮助弥合这些联系。更详细地说，该项目重点介绍了两个Keystone应用程序。第一个是处理轨迹，它可以被建模为空间域中的时间参数曲线，并对在典型分析管道中直接使用提出了许多挑战。开发的方法展示了如何将这些对象(以及相关的低维几何结构)转换为矢量化表示，从而可以与大量数据分析任务无缝集成。第二个是分析已经嵌入高维空间中的点的数据集。在单个数据点坐标没有意义的情况下，单词向量表示是一个中心和重要的这样的例子。因此，该项目使用这些高维数据集的几何形状来提供更丰富的分析操作和更直观、更透明的分析。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.