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 $\mu^2 = -1$
posseses 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 produce a method to bound the spectral norm of a
Quaternionic Convolution.

本论文研究四元数傅里叶变换和卷积运算的矩阵形式及其与标准离散傅里叶变换之间的关系，探讨四元数傅里叶变换矩阵与四元数循环矩阵（代表四元数卷积）的特征结构，给出了一个基于理论结果的方法来限制四元数卷积的谱范数。

四元数傅里叶变换及四元数卷积的矩阵形式

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

Understanding the fundamental principles behind the massive success of neural
networks is one of the most important open questions in deep learning. However,
due to the highly complex nature of the problem, progress has been relatively
slow. In this note, through the lens of infinite-width networks, a.k.a. neural
kernels, we present one such principle resulting from hierarchical localities.
It is well-known that the eigenstructure of infinite-width multilayer
perceptrons (MLPs) depends solely on the concept frequency, which measures the
order of interactions. We show that the topologies from deep convolutional
networks (CNNs) restructure the associated eigenspaces into finer subspaces. In
addition to frequency, the new structure also depends on the concept space,
which measures the spatial distance among nonlinear interaction terms. The
resulting fine-grained eigenstructure dramatically improves the network's
learnability, empowering them to simultaneously model a much richer class of
interactions, including Long-Range-Low-Frequency interactions,
Short-Range-High-Frequency interactions, and various interpolations and
extrapolations in-between. Additionally, model scaling can improve the
resolutions of interpolations and extrapolations and, therefore, the network's
learnability. Finally, we prove a sharp characterization of the generalization
error for infinite-width CNNs of any depth in the high-dimensional setting. Two
corollaries follow: (1) infinite-width deep CNNs can break the curse of
dimensionality without losing their expressivity, and (2) scaling improves
performance in both the finite and infinite data regimes.

通过无限宽度的卷积神经网络理论，研究神经网络的本质原理和泛化能力，提出一种基于层次局部性的新原则并证明了其对神经网络学习能力的提升，同时证明了无限宽度的深度卷积神经网络可以打破维数灾难并保持表达能力，并在有限和无限数据情况下提高性能。