Related papers: On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution

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

URL: http://arxiv.org/abs/2307.01836v3
Date: Mon, 22 Jul 2024 17:29:58 GMT
Title: On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution
Authors: Giorgos Sfikas, George Retsinas,
Abstract summary: 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.
Score: 6.635903943457569
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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}.

Related papers

Machine Learning Mutation-Acyclicity of Quivers [0.0]
This paper applies machine learning techniques to the study of quivers--a type of directed multigraph with significant relevance in algebra. We focus on determining the mutation-acyclicity of a quiver on 4 vertices, a property that is pivotal since mutation-acyclicity is often a necessary condition for theorems involving path algebras and cluster algebras. By neural networks (NNs) and support vector machines (SVMs), we accurately classify more general 4-x quivers as mutation-acyclic or non-mutation-acyclic.
arXiv Detail & Related papers (2024-11-06T19:08:30Z)
Efficient conversion from fermionic Gaussian states to matrix product states [48.225436651971805]
We propose a highly efficient algorithm that converts fermionic Gaussian states to matrix product states. It can be formulated for finite-size systems without translation invariance, but becomes particularly appealing when applied to infinite systems. The potential of our method is demonstrated by numerical calculations in two chiral spin liquids.
arXiv Detail & Related papers (2024-08-02T10:15:26Z)
Exact Correlation Functions for Dual-Unitary Quantum circuits with exceptional points [0.0]
We give an inverse approach for constructing dual-unitary quantum circuits with exceptional points. As a consequence of eigenvectors, correlation functions exhibit a modified exponential decay. We show that behaviors of correlation functions are distinct by Latemporalplace transformation.
arXiv Detail & Related papers (2024-06-12T15:44:29Z)
Entrywise error bounds for low-rank approximations of kernel matrices [55.524284152242096]
We derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues. We validate our theory with an empirical study of a collection of synthetic and real-world datasets.
arXiv Detail & Related papers (2024-05-23T12:26:25Z)
Biquaternion representation of the spin one half and its application on the relativistic one electron atom [65.268245109828]
In this work we represent the $1/2$ Spin particles with complex quaternions. We determine the states, rotation operators and the total angular momentum function in the complex quaternion space.
arXiv Detail & Related papers (2024-02-28T19:24:13Z)
Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
arXiv Detail & Related papers (2023-02-27T18:56:40Z)
Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces [52.424621227687894]
We introduce a unified framework for group equivariant networks on homogeneous spaces. We take advantage of the sparsity of Fourier coefficients of the lifted feature fields. We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation.
arXiv Detail & Related papers (2022-06-16T17:59:01Z)
Transformer with Fourier Integral Attentions [18.031977028559282]
We propose a new class of transformers in which the dot-product kernels are replaced by the novel generalized Fourier integral kernels. Compared to the conventional transformers with dot-product attention, FourierFormers attain better accuracy and reduce the redundancy between attention heads. We empirically corroborate the advantages of FourierFormers over the baseline transformers in a variety of practical applications including language modeling and image classification.
arXiv Detail & Related papers (2022-06-01T03:06:21Z)
Convolutional Filtering and Neural Networks with Non Commutative Algebras [153.20329791008095]
We study the generalization of non commutative convolutional neural networks. We show that non commutative convolutional architectures can be stable to deformations on the space of operators.
arXiv Detail & Related papers (2021-08-23T04:22:58Z)
Conformal bridge between asymptotic freedom and confinement [0.0]
We construct a nonunitary transformation that relates a given "asymptotically free" conformal quantum mechanical system $H_f$ with its confined, harmonically trapped version $H_c$. We investigate the one- and two-dimensional examples that reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.
arXiv Detail & Related papers (2019-12-26T02:45:37Z)

This list is automatically generated from the titles and abstracts of the papers in this site.