One develops a fast computational methodology for principal component analysis on manifolds. Instead of estimating intrinsic principal components on an object space with a Riemannian structure, one embeds the object space in a numerical space, and the resulting chord distance is used. This method helps us analyzing high, theoretically even infinite dimensional data, from a new perspective. We define the extrinsic principal sub-manifolds of a random object on a Hilbert manifold embedded in a Hilbert space, and the sample counterparts. The resulting extrinsic principal components are useful for dimension data reduction. For application, one retains a very small number of such extrinsic principal components for a shape of contour data sample, extracted from imaging data.

翻译：本文提出了一种在流形上进行主成分分析的快速计算方法。该方法并非在具有黎曼结构的对象空间中估计内蕴主成分，而是将对象空间嵌入数值空间，并利用由此产生的弦距离。这种方法有助于我们从新视角分析高维乃至理论上无限维的数据。我们定义了嵌入希尔伯特空间的希尔伯特流形上随机对象的外源性主子流形及其样本对应物。所得的外源性主成分可用于数据降维。在应用方面，针对从成像数据中提取的轮廓数据样本形状，仅需保留极少量的此类外源性主成分。