  It is a known fact that training recurrent neural networks for tasks that
have long term dependencies is challenging. One of the main reasons is the
vanishing or exploding gradient problem, which prevents gradient information
from propagating to early layers. In this paper we propose a simple recurrent
architecture, the Fourier Recurrent Unit (FRU), that stabilizes the gradients
that arise in its training while giving us stronger expressive power.
Specifically, FRU summarizes the hidden states $h^{(t)}$ along the temporal
dimension with Fourier basis functions. This allows gradients to easily reach
any layer due to FRU's residual learning structure and the global support of
trigonometric functions. We show that FRU has gradient lower and upper bounds
independent of temporal dimension. We also show the strong expressivity of
sparse Fourier basis, from which FRU obtains its strong expressive power. Our
experimental study also demonstrates that with fewer parameters the proposed
architecture outperforms other recurrent architectures on many tasks.
