Deep Extrinsic Manifold Representation for Vision Tasks
This addresses a fundamental problem in machine learning for handling non-Euclidean data in vision tasks, though it appears incremental as it builds on existing manifold embedding techniques.
The paper tackles the challenge of training neural networks to output manifold representations for non-Euclidean data by introducing Deep Extrinsic Manifold Representation (DEMR), which embeds manifolds into Euclidean space to avoid complex geodesic losses, and demonstrates its effectiveness on point cloud alignment and illumination subspace learning tasks.
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.