Beyond Riemannian Geometry in Inference

现代数据科学的一个挑战是如何将来自三维形状的信息集成到统计模型中。这一挑战的核心应用例子是将肿瘤的形状与分子过程联系起来，或者将水稻植株的根的形状与作物产量联系起来。在图形学和解剖学中，有一个相关的问题，即如何将一个物体弯曲到另一个物体上，比如儿童的磨牙和成人的磨牙。这个项目寻求开发方法，将这些形状数据，如网格或三维图像，转换为标准统计模型可用的表示法。这些方法对于推动数据使能科学在提取、概念化、解释和可视化驻留在包括3D形状等复杂对象的数据集中的信息方面至关重要。数学中的前沿思想，特别是几何领域的前沿思想，将用于解决(I)对不同形状集合中的结构变化进行建模和(Ii)对形状之间的转换进行建模的基本问题。这些问题的解决对于包括生物学、医学、社会科学和生态学在内的许多实际应用和学科至关重要。因此，该项目的一个重要组成部分是通过在放射学和人类学中的应用来验证方法和工具，将利用黎曼几何和光滑流形以外的几何概念来开发新的统计方法，以应对数据分析中的复杂挑战。将开发以坚实的数学基础为基础的方法和工具，用于对复杂对象(如形状和表面)和数据中的复杂关系(如多种商品流动或对齐形状)进行建模。这项研究将使用几何工具解决两个统计挑战：(1)通过积分几何来表示表面和形状，(2)使用纤维束的几何来学习对象对之间的群体动作或变换。解决第一个挑战导致了用于形状和表面集合的参数和非参数统计模型的框架，而不需要地标点，也不要求形状同构。解决第二个挑战为对齐问题提供了一个统计框架，范围从对齐一组形状(如牙齿)到网络上的优化问题(如多商品流动)。提出的解决方案将对统计学和几何学产生影响，而这些方法可能会在生物、医学、社会科学和生态学等不同的实际应用和科学学科中催化变革性的进步。

A challenge in modern data science is how to integrate information from 3-dimensional shapes into statistical models. Examples of applications where this challenge is central is associating the shape of a tumor to molecular processes or associating the shape of the roots of a rice plant to crop yield. In graphics and anatomy, there is the related question of how to warp one object into another, such as the molar of a child to a molar of an adult. This project seeks to develop methodology to transform these shape data, such as meshes or 3-dimensional images, into representations for which standard statistical models are available. These methods are crucial to advancing data-enabled science in extracting, conceptualizing, interpreting, and visualizing information residing in datasets comprising complex objects such 3D shapes. Cutting edge ideas in mathematics, specifically from the field of geometry, will be used to address the fundamental problems of (i) modeling structural variation in diverse collections of shapes and (ii) modeling transformations between shapes. Solutions to these problems are crucial to many practical applications and disciplines including biology, medicine, social sciences, and ecology. As such, an important component of the project is validation of methods and tools through applications in radiology and anthropologyGeometric concepts beyond Riemannian geometry and smooth manifolds will be leveraged to develop novel statistical methodology to address complex challenges in data analysis. Methods and tools, grounded on solid mathematical foundations, will be developed for modeling complex objects (such as shapes and surfaces) and complex relations within data (such as multi-commodity flow or aligning shapes). The research will address two statistical challenges using geometric tools: (1) representing surfaces and shapes via integral geometry, and (2) learning group actions or transformations between pairs of objects using the geometry of fiber bundles. Addressing the first challenge results in a framework for parametric and non-parametric statistical models for collections of shapes and surfaces, without the requirement of landmark points and without requiring the shapes to be isomorphic. Addressing the second challenge provides a statistical framework for alignment problems ranging from aligning a collection of shapes such as teeth, to optimization problems on networks such as multi-commodity flow. The solutions proposed will impact statistics and geometry, and the methods may catalyze transformative advances in practical applications and scientific disciplines as diverse as biology, medicine, social sciences, and ecology.