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We introduce a novel approach to perform first-order optimization withorthogonal and unitary constraints. This approach is based on a parametrizationstemming from Lie group theory through the exponential map. The parametrizationtransforms the constrained optimization problem into an unconstrained one overa Euclidean space, for which common first-order optimization methods can beused. The theoretical results presented are general enough to cover the specialorthogonal group, the unitary group and, in general, any connected compact Liegroup. We discuss how this and other parametrizations can be computedefficiently through an implementation trick, making numerically complexparametrizations usable at a negligible runtime cost in neural networks. Inparticular, we apply our results to RNNs with orthogonal recurrent weights,yielding a new architecture called expRNN. We demonstrate how our methodconstitutes a more robust approach to optimization with orthogonal constraints,showing faster, accurate, and more stable convergence in several tasks designedto test RNNs.

Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group