发表机构
National Institute of Standards and Technology（美国国家标准与技术研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究如何将活动子空间原理从欧几里得空间推广到黎曼流形，通过与平行传输结合，对比内在与外在方法，在局部意义上研究两者差异，以超球面为例进行数值说明，包括曲率限制下的岭恢复。
AI 中文摘要
活动子空间为研究感兴趣的标量值量如何在欧几里得域的简化基上平均变化最大提供了一种可解释的、按特征值排序的原理。与平行传输相结合将该原理从欧几里得空间推广到黎曼流形上定义的感兴趣的量，所得的内在公式与基于嵌入的流形学习梯度平均值形成对比。在内在局部意义上研究这两种策略，限制在均值中心测地球内，在此范围内两者不同：在中心切空间上，特征值在采样域的测地半径上二阶一致，而主导特征空间相对于谱隙在相同阶上一致。将活动扩展到该中心空间之外需要在变化的切空间上重新计算分解，或者内在地对单个中心框架进行平行传输。超球面作为感兴趣的特定流形贯穿始终，这是由统计形状分析的预形状空间上的应用所推动的。在二维球面上的数值示例说明了该形式主义，包括以曲率限制的二次速率导出的岭恢复。
英文摘要
Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.
