Related papers: The HR-Calculus: Enabling Information Processing w…
This article considers the problem of designing adaption and optimisation techniques for training quantum learning machines. To this end, the division algebra of quaternions is used to derive an effective model for representing computation…
Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the…
This paper presents an experimental study on the application of quaternions in several machine learning algorithms. Quaternion is a mathematical representation of rotation in three-dimensional space, which can be used to represent complex…
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…
Quaternion derivatives in the mathematical literature are typically defined only for analytic (regular) functions. However, in engineering problems, functions of interest are often real-valued and thus not analytic, such as the standard…
A novel framework for a unifying treatment of quaternion valued adaptive filtering algorithms is introduced. This is achieved based on a rigorous account of quaternion differentiability, the proposed I-gradient, and the use of augmented…
Convex optimization methods have been extensively used in the fields of communications and signal processing. However, the theory of quaternion optimization is currently not as fully developed and systematic as that of complex and real…
A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of…
Topological phase classifications have been intensively studied via machine-learning techniques where different forms of the training data are proposed in order to maximize the information extracted from the systems of interests. Due to the…
The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions often require the calculation of the gradient and Hessian, however,…
Quaternions, discovered by Sir William Rowan Hamilton in the 19th century, are a significant extension of complex numbers and a profound tool for understanding three-dimensional rotations. This work explores the quaternion's history,…
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…
Quaternion-valued signal processing has received increasing attention recently. One key operation involved in derivation of all kinds of adaptive algorithms is the gradient operator. Although there have been some derivations of this…
The numerical optimization of continuous functions is a fundamental task in many scientific and engineering domains, ranging from mechanical design to training of artificial intelligence models. Among the most effective and widely used…
Sound is a fundamental and rich source of information; playing a key role in many areas from humanities and social sciences through to engineering and mathematics. Sound is more than just data 'signals'. It encapsulates physical, sensorial…
Quaternions are an important tool to describe the orientation of a molecule. This paper considers the use of quaternions in matching two conformations of a molecule, in interpolating rotations, in performing statistics on orientational…
In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings,…
We propose a novel quaternion product unit (QPU) to represent data on 3D rotation groups. The QPU leverages quaternion algebra and the law of 3D rotation group, representing 3D rotation data as quaternions and merging them via a weighted…
Quaternion contains one real part and three imaginary parts, which provided a more expressive hypercomplex space for learning knowledge graph. Existing quaternion embedding models measure the plausibility of a triplet either through…
We propose a perceptual chromatic adaptation transform for white balance that makes use of split-quaternions. The novelty of the present work, which is motivated by a recently developed quantum-like model of color perception, consists at…