Training Without Orthogonalization, Inference With SVD: A Gradient Analysis of Rotation Representations

Abstract

Recent work has shown that removing orthogonalization during training and applying it only at inference improves rotation estimation in deep learning, with empirical evidence favoring 9D representations with SVD projection. However, the theoretical understanding of why SVD orthogonalization specifically harms training, and why it should be preferred over Gram-Schmidt at inference, remains incomplete. We provide a detailed gradient analysis of SVD orthogonalization specialized to $3 \times 3$ matrices and $SO(3)$ projection. Our central result derives the exact spectrum of the SVD backward pass Jacobian: it has rank $3$ (matching the dimension of $SO(3)$) with nonzero singular values $2/(s_i + s_j)$ and condition number $κ= (s_1 + s_2)/(s_2 + s_3)$, creating quantifiable gradient distortion that is most severe when the predicted matrix is far from $SO(3)$ (e.g., early in training when $s_3 \approx 0$). We further show that even stabilized SVD gradients introduce gradient direction error, whereas removing SVD from the training loop avoids this tradeoff entirely. We also prove that the 6D Gram-Schmidt Jacobian has an asymmetric spectrum: its parameters receive unequal gradient signal, explaining why 9D parameterization is preferable. Together, these results provide the theoretical foundation for training with direct 9D regression and applying SVD projection only at inference.