Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural network approximation. We show that this is indeed possible for shape spaces with a special product structure, namely those smoothly approximable by a direct sum of low-dimensional manifolds. Our proposed architecture leverages this structure by separately learning approximations for the low-dimensional factors and a subsequent combination. After developing the approach as a general framework, we apply it to a shape space of triangular surfaces. Here, typical examples of data manifolds are given through datasets of articulated models and can be factorized, for example, by a Sparse Principal Geodesic Analysis (SPGA). We demonstrate the effectiveness of our proposed approach with experiments on synthetic data as well as manifolds extracted from data via SPGA.

翻译：形状空间中数据流形的参数化可利用黎曼几何的丰富工具箱进行计算。然而，这通常伴随着高昂的计算成本，这引发了能否学习高效神经网络近似的问题。我们证明，对于具有特殊乘积结构的形状空间——即那些能够被低维流形直和光滑逼近的空间——这确实是可能的。我们提出的架构通过分别学习低维因子的近似及随后的组合来利用这一结构。将该方法发展为通用框架后，我们将其应用于三角曲面形状空间。在此，典型的数据流形示例由铰接模型数据集给出，并可通过稀疏主测地线分析（SPGA）进行分解。我们通过合成数据以及从数据中通过SPGA提取的流形进行实验，证明了所提出方法的有效性。