AF: EAGER: Collaborative Research: Integration of Computational Geometry and Statistical Learning for Modern Data Analysis

数据分析是计算科学中的一个基本问题，从计算机图形学到地理信息系统，从传感器网络到社会网络，从经济学到生物科学，在广泛的应用领域中无处不在。驱动现代数据分析的两个互补领域是计算几何和统计学习。前者侧重于描述低维几何现象的详细和精确的模型。后者侧重于给定噪声高维数据的模型的鲁棒性或预测性推断。该项目旨在启动这两个领域之间的对话，以几何为中心主题。它们之间更紧密的互动将使这两个领域受益和发展，并可能从根本上改变我们看待和执行数据分析的方式。具体来说，一方面，学习社区中常见的数据类型对传统的计算几何方法提出了一些挑战。将焦点转移到这些挑战和统计学习中心的不确定性建模上，可以拓宽计算几何的范围，并导致对噪声更具鲁棒性的几何算法和模型，并扩展到高维数据分析。另一方面，计算几何已经发展出许多优雅的结构，这些结构通常包含有关底层领域的详细和精确信息。使用这些结构参数化的模型可以产生更丰富、更可解释的统计学习模型和算法，同时对噪声保持鲁棒性和预测性。这个项目是一个多学科的项目，涉及的领域包括计算几何、算法、统计学、微分几何和拓扑学。教育将被纳入这个项目。

Data analysis is a fundamental problem in computational science, ubiquitous in a broad range of application fields, from computer graphics to geographics information system, from sensor networks to social networks, and from economics to biological science. Two complementary fields that have driven modern data analysis are computational geometry and statistical learning. The former focuses on detailed and precise models characterizing low-dimensional geometric phenomena. The latter focuses on robust or predictive inference of models given noisy high-dimensional data. This project aims to initiate a dialog between these two fields with geometry being the central theme. A closer interaction between them will benefit and advance both fields, and can potentially fundamentally change the way we view and perform data analysis. Specifically, on one hand, the type of data common in the learning community poses several challenges for traditional computational geometry methods. The shift of focus to these challenges and the modeling of uncertainty central in statistical learning can broaden the scope of computational geometry, and lead to geometric algorithms and models that are more robust to noise and extend to high-dimensional data analysis. On the other hand, computational geometry has developed many elegant structures that contain often detailed and precise information about the underlying domain. Models parameterized using these structures can lead to statistical learning models and algorithms that are richer and more interpretable but remain robust to noise and are predictive. This project is multi-disciplinary in nature, and will involve fields including computational geometry, algorithms, statistics, differential geometry and topology. Education will be integrated in this project.