A growing number of problems in data analysis and classification involve data that are non-Euclidean. For such problems, a naive application of vector space analysis algorithms will produce results that depend on the choice of local coordinates used to parametrize the data. At the same time, many data analysis and classification problems eventually reduce to an optimization, in which the criteria being minimized can be interpreted as the distortion associated with a mapping between two curved spaces. Exploiting this distortion minimizing perspective, we first show that manifold learning problems involving non-Euclidean data can be naturally framed as seeking a mapping between two Riemannian manifolds that is closest to being an isometry. A family of coordinate-invariant first-order distortion measures is then proposed that measure the proximity of the mapping to an isometry, and applied to manifold learning for non-Euclidean data sets. Case studies ranging from synthetic data to human mass-shape data demonstrate the many performance advantages of our Riemannian distortion minimization framework.

数据分析和分类中越来越多的问题涉及非欧几里得数据。对于此类问题，矢量空间分析算法的简单应用将产生取决于用于参数化数据的局部坐标的选择的结果。同时，许多数据分析和分类问题最终简化为优化，其中最小化的标准可以解释为与两个弯曲空间之间的映射关联的失真。利用这种最小化失真的观点，我们首先表明，涉及非欧几里得数据的流形学习问题可以自然地构图为在两个最接近等轴测图的黎曼流形之间寻求映射。然后提出了一系列坐标不变的一阶失真度量，该度量度量了映射与等轴测图的接近度，并应用于非欧几里得数据集的流形学习。从合成数据到人体质量数据的案例研究表明，我们的黎曼失真最小化框架具有许多性能优势。