Learning Unitary Operators with Help From u(n)

A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie algebra u(n) associated with the Lie group U(n) of n× n unitary matrices. The exponential map provides a correspondence between these spaces, and allows us to define a unitary matrix using n real coefficients relative to a basis of the Lie algebra. The parametrization is closed under additive updates of these coefficients, and thus provides a simple space in which to do gradient descent. We demonstrate the effectiveness of this parametrization on the problem of learning arbitrary unitary operators, comparing to several baselines and outperforming a recently-proposed lower-dimensional parametrization. This suggests a route to generalising a recently-proposed unitary recurrent neural network to arbitrary unitary matrices, solving a problem the well-known long short-term memory network was invented to address, but with a simplified and elegant network architecture.