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

TL;DR

The paper develops a matrix-based framework for the Quaternion Fourier Transform and Quaternion Convolution, clarifying their relationship to the complex DFT and quaternion circulant matrices. It shows that an infinite family of Quaternionic Fourier matrices generalizes the standard DFT and that quaternionic circulant structures diagonalize under these transforms via left/right spectra, including a 2D extension with doubly-block circulant matrices. A key contribution is a fast, geometry-aware method to bound the Lipschitz constant of quaternionic neural networks by exploiting left eigenvalues through a structured QSVD-based singular value clipping, avoiding explicit construction of large doubly-block circulant matrices. The results enable efficient spectral analysis and reconstruction in quaternionic signal processing, with practical benefits demonstrated on CIFAR-10 using quaternionic ResNets and publicly available code.

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 $μ^2 = -1$ 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}.

Figures (6)

• Figure 1: Per-epoch analysis of the impact of using the proposed technique. Training loss (top) and test accuracy (botoom) curves are reported.
• Figure 2: Distribution of the singular values for each layers, when training a QResNet32-small architecture. The reported values are for the final model, after 80 epochs.
• Figure 3: Test accuracy curves during the training for the QResNet32-small architecture. Four randomly initialized $\mu$ values are considered for a clipping value of $c=0.5$. Baseline w/o clipping is reported as a dashed line at 82.77%.
• Figure 4: Constraining the spectral norm using QSVD directly on the doubly-block circulant matrix (referred to as "brute-force" in text) vs our method. From top to bottom, we see: the original kernel, the clipped kernel using our method, the point-to-point difference between our result and the brute-force method. From left to right, we see log-magnitudes, phases and axes of respective doubly-block circulant matrices (cf. Section \ref{['sec:preliminaries']} of the Appendix, or ell2007hypercomplex). The brute-force method is considerably more expensive in terms of space as well as time requirements, both by orders of magnitude. For this $32 \times 32$ kernel, our method required $280$ milliseconds to obtain an exact reconstruction, while brute-force took over $2$ minutes. (see text of Section \ref{['sec:Lipschitz']} for more details).
• Figure 5: Distribution of the singular values for each layers, when training a QResNet32-small architecture. The reported values are for the final model, after 80 epochs.

Theorems & Definitions (20)

• Proposition 3.1
• Proposition 3.2
• Corollary 3.2.1
• Proposition 3.3: General properties
• Proposition 3.4
• Proposition 3.5: Circulant & Fourier Matrices
• Corollary 3.5.1
• Corollary 3.5.2
• Proposition 3.6
• Corollary 3.6.1