Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform
This work addresses a computational bottleneck for researchers and practitioners in deep learning who need to enforce orthonormality constraints efficiently.
The paper tackles the computational expense of Riemannian optimization on the Stiefel manifold by introducing new retraction and vector transport methods based on the Cayley transform, resulting in algorithms that use less running time per iteration and achieve faster convergence rates without compromising performance in CNN and RNN training.
Strictly enforcing orthonormality constraints on parameter matrices has been shown advantageous in deep learning. This amounts to Riemannian optimization on the Stiefel manifold, which, however, is computationally expensive. To address this challenge, we present two main contributions: (1) A new efficient retraction map based on an iterative Cayley transform for optimization updates, and (2) An implicit vector transport mechanism based on the combination of a projection of the momentum and the Cayley transform on the Stiefel manifold. We specify two new optimization algorithms: Cayley SGD with momentum, and Cayley ADAM on the Stiefel manifold. Convergence of Cayley SGD is theoretically analyzed. Our experiments for CNN training demonstrate that both algorithms: (a) Use less running time per iteration relative to existing approaches that enforce orthonormality of CNN parameters; and (b) Achieve faster convergence rates than the baseline SGD and ADAM algorithms without compromising the performance of the CNN. Cayley SGD and Cayley ADAM are also shown to reduce the training time for optimizing the unitary transition matrices in RNNs.