  Recurrent neural networks are extremely powerful yet hard to train. One of
their issues is the vanishing gradient problem, whereby propagation of training
signals may be exponentially attenuated, freezing training. Use of orthogonal
or unitary matrices, whose powers neither explode nor decay, has been proposed
to mitigate this issue, but their computational expense has hindered their use.
Here we show that in the specific case of convolutional RNNs, we can define a
convolutional exponential and that this operation transforms antisymmetric or
anti-Hermitian convolution kernels into orthogonal or unitary convolution
kernels. We explicitly derive FFT-based algorithms to compute the kernels and
their derivatives. The computational complexity of parametrizing this subspace
of orthogonal transformations is thus the same as the networks' iteration.
