Clive Cheong Took

Min Xiang, Shirin Enshaeifar, Alexander E. Stott et al.

Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the non-commutative nature of the quaternion product has been prohibitive to the development of quaternion uncorrelating transforms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance and complementary covariance matrices in the quaternion domain, whereby the quaternion version of the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.

Danilo P. Mandic, Sayed Pouria Talebi, Clive Cheong Took et al.

From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.

Sayed Pouria Talebi, Clive Cheong Took

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 success of quaternions. The most useful feature of quaternions lies in their ability to model three-dimensional rotations which, in turn, have found various industrial applications such as in aeronautics and computergraphics. Recently, we have witnessed a renaissance of quaternions due to the rise of machine learning. To equip the reader to contribute to this emerging research area, this chapter lays down the foundation for: - augmented statistics for modelling quaternion-valued random processes, - widely linear models to exploit such advanced statistics, - quaternion calculus and algebra for algorithmic derivations, - mean square estimation for practical considerations. For ease of exposure, several examples are offered to facilitate the learning, understanding, and(hopefully) the adoption of this multidimensional domain.

Sayed Pouria Talebi, Clive Cheong Took, Danilo P. Mandic

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 and measurement operations on qubits. In turn, the derived model, serves as the foundation for formulating an adaptive learning problem on principal quantum learning units, thereby establishing quantum information processing units akin to that of neurons in classical approaches. Then, leveraging the modern HR-calculus, a comprehensive training framework for learning on quantum machines is developed. The quaternion-valued model accommodates mathematical tractability and establishment of performance criteria, such as convergence conditions.

Nikhil Banerjee, Thanassis Giannetsos, Emmanouil Panaousis et al.

The advancement of Internet-of-Things (IoT) edge devices with various types of sensors enables us to harness diverse information with Mobile Crowd-Sensing applications (MCS). This highly dynamic setting entails the collection of ubiquitous data traces, originating from sensors carried by people, introducing new information security challenges; one of them being the preservation of data trustworthiness. What is needed in these settings is the timely analysis of these large datasets to produce accurate insights on the correctness of user reports. Existing data mining and other artificial intelligence methods are the most popular to gain hidden insights from IoT data, albeit with many challenges. In this paper, we first model the cyber trustworthiness of MCS reports in the presence of intelligent and colluding adversaries. We then rigorously assess, using real IoT datasets, the effectiveness and accuracy of well-known data mining algorithms when employed towards IoT security and privacy. By taking into account the spatio-temporal changes of the underlying phenomena, we demonstrate how concept drifts can masquerade the existence of attackers and their impact on the accuracy of both the clustering and classification processes. Our initial set of results clearly show that these unsupervised learning algorithms are prone to adversarial infection, thus, magnifying the need for further research in the field by leveraging a mix of advanced machine learning models and mathematical optimization techniques.