Related papers: The geometry of proper quaternion random variables

This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large $p$, large $n$" scenario, where $p$ represents the number of variables and $n$ is the sample size. Our analysis…

Extensions of real numbers in more than two dimensions, in particular quaternions and octonions are finding applications in physics due to the fact that they naturally capture certain symmetries of physical systems. Here it is shown that…

Dimensionality reduction is a crucial technique in data analysis, as it allows for the efficient visualization and understanding of high-dimensional datasets. The circular coordinate is one of the topological data analysis techniques…

Quaternion symmetry is ubiquitous in the physical sciences. As such, much work has been afforded over the years to the development of efficient schemes to exploit this symmetry using real and complex linear algebra. Recent years have also…

Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…

Quaternions provide a unified algebraic and geometric framework for representing three-dimensional rotations without the singularities that afflict Euler-angle parametrisations. This article develops a pedagogical and conceptual analysis of…

The paper explains how a unit generalized quaternion is used to represent a rotation of a vector in 3-dimensional space. We review of some algebraic properties of generalized quaternions and operations between them and then show their…

We prove that quaternion Gaussian random matrices satisfy the restricted isometry property (RIP) with overwhelming probability. We also explain why the restricted isometry random variables (RIV) approach is not appropriate for drawing…

In this paper we extend some previous results for a new proposal for gravity and place them into overall context. The basic fields of this proposal provide an off-shell realization of symmetry with respect to SO(3,C) gauge transformations…

For quaternionic signal processing algorithms, the gradients of a quaternion-valued function are required for gradient-based methods. Given the non-commutativity of quaternion algebra, the definition of the gradients is non-trivial. The HR…

After introducing the definitions of positive, negative and companion rules, from a given pair of companion rules we construct a new rule with higher degree of precision The scheme is generalized giving rise to a transformation which we…

A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…

The purpose of this paper is to propose a new multi-layer feedforward quaternion neural network model architecture, Reverse Quaternion Neural Network which utilizes the non-commutative nature of quaternion products, and to clarify its…

This paper focuses on regularisation methods using models up to the third order to search for up to second-order critical points of a finite-sum minimisation problem. The variant presented belongs to the framework of [3]: it employs random…

The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for…

We consider quadrangles of perimeter $2$ in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^\bot$ defines a plane…

The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere…

In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing…

Quaternion analysis is considered in full details where a new analyticity condition in complete analogy to complex analysis is found. The extension to octonions is also worked out.

Quaternion (Q-) mathematics formally contains many fragments of physical laws; in particular, the Hamiltonian for the Pauli equation automatically emerges in a space with Q-metric. The eigenfunction method shows that any Q-unit has an…