Convolutional unitary or orthogonal recurrent neural networks

TL;DR

This paper introduces a computationally efficient method for convolutional RNNs to use orthogonal or unitary kernels via a convolutional exponential, addressing training difficulties like vanishing gradients.

Contribution

It defines a convolutional exponential to transform antisymmetric kernels into orthogonal or unitary ones, with FFT-based algorithms for efficient computation.

Findings

FFT-based algorithms enable efficient kernel computation.

The method maintains the same complexity as standard network iterations.

It effectively mitigates vanishing gradient issues in convolutional RNNs.

Abstract

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.