Non-Euclidean data is frequently encountered across different fields, yet
there is limited literature that addresses the fundamental challenge of
training neural networks with manifold representations as outputs. We introduce
the trick named Deep Extrinsic Manifold Representation (DEMR) for visual tasks
in this context. DEMR incorporates extrinsic manifold embedding into deep
neural networks, which helps generate manifold representations. The DEMR
approach does not directly optimize the complex geodesic loss. Instead, it
focuses on optimizing the computation graph within the embedded Euclidean
space, allowing for adaptability to various architectural requirements. We
provide empirical evidence supporting the proposed concept on two types of
manifolds, $SE(3)$ and its associated quotient manifolds. This evidence offers
theoretical assurances regarding feasibility, asymptotic properties, and
generalization capability. The experimental results show that DEMR effectively
adapts to point cloud alignment, producing outputs in $ SE(3) $, as well as in
illumination subspace learning with outputs on the Grassmann manifold.

通过将外在流形表征嵌入到深度神经网络中，Deep Extrinsic Manifold Representation (DEMR) 方法用于视觉任务，不直接优化复杂的测地损失，而是侧重于优化嵌入欧氏空间内的计算图，以适应不同的架构要求；在两种流形类型 ——$SE (3)$ 及其相关商流形上，提供了理论上的可行性、渐近性质和泛化能力的实验证据。DEMR 在点云对齐和光照子空间学习方面表现出良好的适应性。