Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis
English summary
This survey synthesizes research on shape space analysis, a mathematical framework for studying geometric data arising in biology, medicine, anthropology, and computer vision. The authors organize the literature around an analytical pipeline: shape representation and parameterization, construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. Applications span from subcellular morphology to primate tooth evolution, demonstrating how the framework captures nonlinear geometric variations. The review identifies key theoretical and computational challenges and highlights emerging opportunities with large geometric datasets.
Chinese summary
该综述综合了形状空间分析的研究，这是一个用于研究生物学、医学、人类学和计算机视觉等领域几何数据的数学框架。作者围绕分析流程组织文献：形状表示与参数化、稳健测地度量的构建、形状空间上的统计分析以及几何感知学习方法。应用范围从亚细胞形态到灵长类牙齿进化，展示了该框架如何捕捉非线性几何变异。综述指出了关键的理论和计算挑战，并突出了大规模几何数据集带来的新机遇。
Key points
Shape space analysis provides a mathematical and computational framework for studying geometric data with nonlinear structures.
形状空间分析为研究具有非线性结构的几何数据提供了数学和计算框架。
The literature is organized into a pipeline: shape representation, geodesic metrics, statistical shape analysis, and geometry-aware learning.
文献被组织为一个流程：形状表示、测地度量、统计形状分析和几何感知学习。
Showcases cross-disciplinary applications from subcellular biology to evolutionary anthropology, tackling complex geometric variations.
展示了从亚细胞生物学到进化人类学的跨学科应用，处理复杂的几何变异。
Identifies future directions including theoretical challenges and opportunities from large-scale geometric datasets.
指明了未来方向，包括理论挑战和大规模几何数据集带来的机遇。