We propose extrinsic and intrinsic deep neural network architectures as general frameworks for deep learning on manifolds. Specifically, extrinsic deep neural networks (eDNNs) preserve geometric features on manifolds by utilizing an equivariant embedding from the manifold to its image in the Euclidean space. Moreover, intrinsic deep neural networks (iDNNs) incorporate the underlying intrinsic geometry of manifolds via exponential and log maps with respect to a Riemannian structure. Consequently, we prove that the empirical risk of the empirical risk minimizers (ERM) of eDNNs and iDNNs converge in optimal rates. Overall, The eDNNs framework is simple and easy to compute, while the iDNNs framework is accurate and fast converging. To demonstrate the utilities of our framework, various simulation studies, and real data analyses are presented with eDNNs and iDNNs.

翻译：我们提出了外在和内在深度神经网络架构，作为流形上深度学习的通用框架。具体而言，外在深度神经网络（eDNNs）通过利用从流形到其欧几里得空间像的等变嵌入，保留流形上的几何特征。此外，内在深度神经网络（iDNNs）通过基于黎曼结构的指数映射和对数映射，纳入流形的内在几何信息。由此，我们证明了eDNNs和iDNNs的经验风险最小化器（ERM）的经验风险以最优速率收敛。总体而言，eDNNs框架简单且易于计算，而iDNNs框架精确且收敛快速。为展示我们框架的实用性，我们基于eDNNs和iDNNs进行了多项模拟研究和实际数据分析。