On the Continuity of Rotation Representations in Neural Networks (1812.07035v4)

Abstract: In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this to topological concepts such as homeomorphism and embedding. We then investigate what are continuous and discontinuous representations for 2D, 3D, and n-dimensional rotations. We demonstrate that for 3D rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions. Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. We also present continuous representations for the general case of the n-dimensional rotation group SO(n). While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms. We finally present empirical results, which show that our continuous rotation representations outperform discontinuous ones for several practical problems in graphics and vision, including a simple autoencoder sanity test, a rotation estimator for 3D point clouds, and an inverse kinematics solver for 3D human poses.

• The paper establishes that continuous 5D and 6D rotation representations enable smoother neural network training.
• It demonstrates that traditional representations like Euler angles and quaternions suffer from inherent topological discontinuities.
• Empirical tests reveal that the proposed methods significantly reduce errors in autoencoders, pose estimation, and inverse kinematics tasks.

An In-Depth Analysis of "On the Continuity of Rotation Representations in Neural Networks"

In "On the Continuity of Rotation Representations in Neural Networks," Yi Zhou et al. delineate the nuances and challenges associated with representing rotations in the context of deep learning. The authors embark on a methodical exploration of the topological properties of various rotation representations to determine their suitability for neural network training. This paper is pivotal as it provides a rigorous theoretical framework and empirical evidence demonstrating the impact of continuity on the performance of neural networks tasked with learning rotation representations.

The Core Argument

The paper starts by introducing the fundamental problem: neural networks often struggle with learning rotations due to discontinuities inherent in common representations such as Euler angles and quaternions. The authors define continuity in the context of neural networks and connect it to topological concepts like homeomorphism and embedding. They establish that continuous representations are generally more conducive to network training, as smoother functions can be approximated more easily and accurately by neural networks.

Continuity in Rotation Representations

Zhou et al. present a detailed theoretical analysis demonstrating that commonly used 3D and 4D representations of rotations (e.g., quaternions, axis-angle, and Euler angles) are inherently discontinuous when the full rotation space is required. Specifically, they show that for 3D rotations, all conventional representations are discontinuous when mapped to a four or fewer dimensional Euclidean space. This discontinuity poses learning challenges for neural networks, as it can lead to significant approximation errors at certain rotation angles.

Proposed Solutions

To address these challenges, the authors introduce new continuous representations for rotations. They demonstrate that 3D rotations can be represented continuously in 5D and 6D spaces. The 6D representation is based on a Gram-Schmidt-like process that ensures orthogonality, while the 5D representation further reduces dimensionality using normalized projections. These representations maintain continuity, thus facilitating smoother training processes for neural networks.

Empirical Validation

The authors validate their theoretical insights through a series of empirical tests. They implement three primary experiments:

1. Sanity Test with Autoencoders: They demonstrate that networks trained with their 5D and 6D continuous representations converge faster and achieve lower errors compared to those trained with traditional discontinuous representations. The errors from networks using discontinuous representations can be up to 180 degrees, showcasing the stark contrast.
2. Pose Estimation for 3D Point Clouds: Using a simplified PointNet architecture, they show that their continuous representations lead to more accurate and stable rotation estimates for point clouds. Specifically, the continuous 6D representation results in significantly lower mean and maximum errors.
3. Inverse Kinematics for 3D Human Poses: The authors illustrate the practical implications of their research by training a network to solve inverse kinematics problems. They find that networks using the 6D representation yield lower joint position errors compared to those using quaternions or other common representations.

Implications and Future Directions

This paper's contributions have both practical and theoretical implications. Practically, the continuous 5D and 6D rotation representations can be directly employed in various graphics and vision tasks, including pose estimation, motion capture, and robotic kinematics, where rotations play a critical role. Theoretically, the results advocate for a reassessment of how topological properties, such as continuity, influence the design and training of neural networks for geometric problems.

Moving forward, the research opens avenues for exploring continuous representations in other domains and groups beyond rotations, such as similarity transforms and orthogonal groups. Additionally, it prompts further investigation into the relationship between neural network architecture and the topological properties of the data representation space.

Conclusion

In summary, "On the Continuity of Rotation Representations in Neural Networks" by Yi Zhou et al. provides a comprehensive analysis of the topological challenges in rotation representations and offers practical solutions to improve the training of neural networks. By establishing continuity as a critical factor for effective learning, the paper lays the groundwork for future research in both theoretical and applied machine learning contexts.