On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution

TL;DR

This paper explores the matrix representation of quaternion Fourier transforms and convolutions, clarifying their relation to complex Fourier matrices and eigenstructures, with applications to bounding Lipschitz constants in quaternionic neural networks.

Contribution

It provides a detailed analysis of quaternion Fourier transform matrices, their relation to complex Fourier matrices, and their eigenstructure, along with a practical method for neural network Lipschitz bounds.

Findings

Quaternion Fourier Transform matrices relate closely to complex Fourier matrices.

Eigenstructure of quaternionic circulant matrices is characterized.

A method to bound Lipschitz constants in quaternionic CNNs is demonstrated.

Abstract

We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from non-commutativity of quaternion multiplication, and due to that $\mu^2 = -1$ possesses infinite solutions in the quaternion domain. Handling of quaternionic matrices is consequently complicated in several aspects (definition of eigenstructure, determinant, etc.). Our research findings clarify the relation of the Quaternion Fourier Transform matrix to the standard (complex) Discrete Fourier Transform matrix, and the extend on which well-known complex-domain theorems extend to quaternions. We focus especially on the relation of Quaternion Fourier Transform matrices to Quaternion Circulant matrices (representing quaternionic convolution), and the eigenstructure of the latter. A proof-of-concept application that makes direct use of our theoretical results is presented, where we present a method to bound the Lipschitz constant of a Quaternionic Convolutional Neural Network. Code is publicly available at: \url{https://github.com/sfikas/quaternion-fourier-convolution-matrix}.