CWY Parametrization for Scalable Learning of Orthogonal and Stiefel Matrices

In this paper we propose a new approach for optimization over orthogonal groups. We parametrize an orthogonal matrix as a product of Householder reflections. To overcome low parallelization capabilities of computing Householder reflections sequentially, we employ an accumulation scheme called the compact WY (or CWY) transform---a compact matrix representation for the series of Householder reflections which can be computed efficiently on highly parallelizable computation units such as GPU and TPU. We further introduce the Truncated CWY (or T-CWY)---a novel approach for Stiefel manifold parametrization which has a competitive complexity estimate compared to other methods and, again, has an advantage when computed on GPU and TPU. We apply these proposed parametrizations to train recurrent neural network architectures in the tasks of neural machine translation and video prediction and demonstrate superiority in both computational and learning aspects compared to other methods from the literature.