如果你处理非线性的几何数据（比如生物形态或3D模型），这篇综述总结了形状空间分析的核心方法，帮你理解如何比较和统计分析几何对象。
这篇综述整理了形状空间分析的快速发展领域，提供了基于微分几何、统计学和机器学习的数学与计算框架。它围绕形状表示、鲁棒测地度量构建、形状空间统计分析和几何感知学习方法组织文献。应用涵盖亚细胞形态学和灵长类牙齿演化等多个生物组织尺度。文章最后指出了理论计算挑战以及由大规模几何数据集驱动的新机遇。
Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis
A central objective of machine learning is to identify structure and patterns in data. Advances in data acquisition have increasingly produced datasets whose observations possess rich geometric form, giving rise to shape spaces that encode variability in object geometry. Such datasets arise across a wide range of disciplines, including biology, medicine, anthropology, and computer vision, where subtle geometric differences often carry important scientific information. Traditional machine learning methods, however, are frequently ill-equipped to account for the nonlinear geometric structure underlying these data. This survey synthesizes a rapidly growing body of work on shape space analysis, which provides a mathematical and computational framework for the study of geometric data. Drawing on ideas from differential geometry, statistics, and machine learning, we organize the literature around a common analytical pipeline: shape representation and parameterization, the rigorous construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. We discuss how these tools enable the characterization of shape variability, the comparison of geometric objects, and the analysis of structural trajectories across populations and time. To illustrate the breadth of the field, we highlight applications spanning multiple scales of biological organization, including studies of subcellular morphology and primate tooth evolution. Across these and many other domains, researchers face common challenges arising from complex, nonlinear, and often unaligned geometric variation. The review concludes by identifying key theoretical and computational challenges, as well as emerging opportunities driven by increasingly large and diverse geometric datasets.