Learning Unitary Operators with Help From u(n)

TL;DR

This paper introduces a Lie algebra-based parametrization for unitary operators in recurrent neural networks, enabling effective gradient-based learning and addressing vanishing/exploding gradient issues.

Contribution

It presents a novel Lie algebra-based parametrization of unitary matrices that simplifies training and improves performance over existing methods.

Findings

Outperforms previous low-dimensional parametrizations.

Successfully learns arbitrary unitary operators.

Effectively solves long-memory tasks with unitary RNNs.

Abstract

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 $\mathfrak{u}(n)$ associated with the Lie group $U(n)$ of $n \times n$ unitary matrices. The exponential map provides a correspondence between these spaces, and allows us to define a unitary matrix using $n^2$ 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. We additionally use our parametrization to generalize a recently-proposed unitary recurrent neural network to arbitrary unitary matrices, using it to solve standard long-memory tasks.