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

In recent years, a deep learning framework has been widely used for object
pose estimation. While quaternion is a common choice for rotation
representation, it cannot represent the ambiguity of the observation. In order
to handle the ambiguity, the Bingham distribution is one promising solution.
However, it requires complicated calculation when yielding the negative
log-likelihood (NLL) loss. An alternative easy-to-implement loss function has
been proposed to avoid complex computations but has difficulty expressing
symmetric distribution. In this paper, we introduce a fast-computable and
easy-to-implement NLL loss function for Bingham distribution. We also create
the inference network and show that our loss function can capture the symmetric
property of target objects from their point clouds.

本文研究物体姿态估计中的旋转表示方式，提出了一种在 Bingham 分布下易于实现和计算的负对数似然损失函数，并证明了其对称性能够良好捕捉目标物体的共性。

一种概率旋转表示对称形状的高效可计算 Bingham Loss 函数

A Probabilistic Rotation Representation for Symmetric Shapes With an  Efficiently Computable Bingham Loss Function

Recently, the connectionist temporal classification (CTC) model coupled with
recurrent (RNN) or convolutional neural networks (CNN), made it easier to train
speech recognition systems in an end-to-end fashion. However in real-valued
models, time frame components such as mel-filter-bank energies and the cepstral
coefficients obtained from them, together with their first and second order
derivatives, are processed as individual elements, while a natural alternative
is to process such components as composed entities. We propose to group such
elements in the form of quaternions and to process these quaternions using the
established quaternion algebra. Quaternion numbers and quaternion neural
networks have shown their efficiency to process multidimensional inputs as
entities, to encode internal dependencies, and to solve many tasks with less
learning parameters than real-valued models. This paper proposes to integrate
multiple feature views in quaternion-valued convolutional neural network
(QCNN), to be used for sequence-to-sequence mapping with the CTC model.
Promising results are reported using simple QCNNs in phoneme recognition
experiments with the TIMIT corpus. More precisely, QCNNs obtain a lower phoneme
error rate (PER) with less learning parameters than a competing model based on
real-valued CNNs.

该研究论文介绍了一种新的方法，使用四元数代替传统的时间帧维度元素作为输入特征来训练基于 QCNN 和 CTC 模型进行序列到序列映射的语音识别系统，实验结果表明，相较于基于传统卷积神经网络的竞争模型，该方法具有更高的准确率且参数更少。