Projective Manifold Gradient Layer for Deep Rotation Regression (2110.11657v3)

Abstract: Regressing rotations on SO(3) manifold using deep neural networks is an important yet unsolved problem. The gap between the Euclidean network output space and the non-Euclidean SO(3) manifold imposes a severe challenge for neural network learning in both forward and backward passes. While several works have proposed different regression-friendly rotation representations, very few works have been devoted to improving the gradient backpropagating in the backward pass. In this paper, we propose a manifold-aware gradient that directly backpropagates into deep network weights. Leveraging Riemannian optimization to construct a novel projective gradient, our proposed regularized projective manifold gradient (RPMG) method helps networks achieve new state-of-the-art performance in a variety of rotation estimation tasks. Our proposed gradient layer can also be applied to other smooth manifolds such as the unit sphere. Our project page is at https://jychen18.github.io/RPMG.

• The paper presents the Regularized Projective Manifold Gradient (RPMG) that improves deep rotation regression by incorporating manifold-aware gradient updates.
• It leverages Riemannian optimization on SO(3) to project gradients onto the manifold, ensuring accurate weight updates and reduced errors.
• Extensive experiments show that RPMG outperforms conventional methods in tasks like 3D pose estimation, advancing rotation estimation in AI.

Overview of the Projective Manifold Gradient Layer for Deep Rotation Regression

Estimating rotations is a fundamental problem in various fields including computer vision, robotics, and graphics, often framed within the context of regression on the special orthogonal group SO(3). This paper presents a novel approach for improving the performance of deep neural networks in rotation regression by addressing the challenges associated with the non-Euclidean nature of SO(3). The authors propose the Regularized Projective Manifold Gradient (RPMG), optimizing both the forward and backward paths of neural networks on manifolds.

Main Contributions

1. Manifold-aware gradient layer: The authors introduce RPMG, a gradient layer that intelligently modifies the backpropagation process in rotation regression. RPMG can be integrated with existing rotation representations, including quaternions, 6D, 9D, and 10D formats, without altering their foundational structures.
2. Riemannian optimization implementation: RPMG leverages Riemannian optimization principles allowing gradients to adhere to the manifold's geometry. The method involves mapping a predicted rotation onto a goal rotation, using steps that respect the manifold's geometry, thus minimizing off-manifold deviations that can introduce error into the training process.
3. Validation across diverse tasks: Extensive experiments demonstrate RPMG's efficacy in various rotation estimation tasks, including 3D object pose estimation from point clouds, rotation estimation without direct supervision, and regression tasks on other manifolds such as the unit sphere. RPMG consistently outperforms standard approaches across these applications.

Technical Details

• Gradient backpropagation challenges: RPMG effectively tackles the discrepancy between Euclidean output spaces of neural networks and the non-Euclidean SO(3) manifold. Previous attempts often neglect the manifold's structure in gradient updates, which RPMG directly manages by using manifold-aware updates.
• Projective manifold gradient: This concept involves projecting gradients derived from Euclidean space back onto the manifold, enabling more accurate weight updates. The RPMG accounts for the geometry of the manifold in determining the steepest descent path.
• Regularization and hyperparameters: RPMG includes a regularization term to maintain output vector norms, preventing their vanishing throughout training. The paper discusses optimal choices for hyperparameters, emphasizing that while RPMG introduces new parameters, their choice is not overburdened by tuning requirements.

Implications and Prospective Developments

The RPMG method provides a robust approach to handling non-Euclidean geometries in neural networks, particularly valuable for tasks involving complex spatial rotations. Practically, it enables advancements in applications such as robotic navigation, augmented reality, and 3D scene understanding where precise rotation estimation is critical.

Theoretically, RPMG highlights the importance of acknowledging differential geometry in designing deep learning models, possibly paving the way for manifold-specific learning paradigms. Future developments in AI could build upon RPMG, crafting more generalized techniques for manifold optimization that could be extended beyond SO(3) to other sophisticated geometries encountered in AI tasks.

Conclusion

The paper contributes significantly to the niche yet critical area of deep rotation regression, providing a novel perspective on gradient optimization within non-Euclidean spaces. RPMG offers substantial improvements in performance across diverse tasks, underscoring the importance of manifold-aware approaches in the ongoing evolution of AI methodologies.