Quaternion recurrent neural network with real-time recurrent learning and maximum correntropy criterion

We develop a robust quaternion recurrent neural network (QRNN) for real-time processing of 3D and 4D data with outliers. This is achieved by combining the real-time recurrent learning (RTRL) algorithm and the maximum correntropy criterion (MCC) as a loss function. While both the mean square error and maximum correntropy criterion are viable cost functions, it is shown that the non-quadratic maximum correntropy loss function is less sensitive to outliers, making it suitable for applications with multidimensional noisy or uncertain data. Both algorithms are derived based on the novel generalised HR (GHR) calculus, which allows for the differentiation of real functions of quaternion variables and offers the product and chain rules, thus enabling elegant and compact derivations. Simulation results in the context of motion prediction of chest internal markers for lung cancer radiotherapy, which includes regular and irregular breathing sequences, support the analysis.

Widely Linear Matched Filter: A Lynchpin towards the Interpretability of Complex-valued CNNs

A recent study on the interpretability of real-valued convolutional neural networks (CNNs) {Stankovic_Mandic_2023CNN} has revealed a direct and physically meaningful link with the task of finding features in data through matched filters. However, applying this paradigm to illuminate the interpretability of complex-valued CNNs meets a formidable obstacle: the extension of matched filtering to a general class of noncircular complex-valued data, referred to here as the widely linear matched filter (WLMF), has been only implicit in the literature. To this end, to establish the interpretability of the operation of complex-valued CNNs, we introduce a general WLMF paradigm, provide its solution and undertake analysis of its performance. For rigor, our WLMF solution is derived without imposing any assumption on the probability density of noise. The theoretical advantages of the WLMF over its standard strictly linear counterpart (SLMF) are provided in terms of their output signal-to-noise-ratios (SNRs), with WLMF consistently exhibiting enhanced SNR. Moreover, the lower bound on the SNR gain of WLMF is derived, together with condition to attain this bound. This serves to revisit the convolution-activation-pooling chain in complex-valued CNNs through the lens of matched filtering, which reveals the potential of WLMFs to provide physical interpretability and enhance explainability of general complex-valued CNNs. Simulations demonstrate the agreement between the theoretical and numerical results.

The HR-Calculus: Enabling Information Processing with Quaternion Algebra

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.

TensorGPT: Efficient Compression of the Embedding Layer in LLMs based on the Tensor-Train Decomposition

High-dimensional token embeddings underpin Large Language Models (LLMs), as they can capture subtle semantic information and significantly enhance the modelling of complex language patterns. However, the associated high dimensionality also introduces considerable model parameters, and a prohibitively high model storage. To address this issue, this work proposes an approach based on the Tensor-Train Decomposition (TTD), where each token embedding is treated as a Matrix Product State (MPS) that can be efficiently computed in a distributed manner. The experimental results on GPT-2 demonstrate that, through our approach, the embedding layer can be compressed by a factor of up to 38.40 times, and when the compression factor is 3.31 times, even produced a better performance than the original GPT-2 model.

A Deep Matched Filter For R-Peak Detection in Ear-ECG

The Ear-ECG provides a continuous Lead I electrocardiogram (ECG) by measuring the potential difference related to heart activity using electrodes that can be embedded within earphones. The significant increase in wearability and comfort afforded by Ear-ECG is often accompanied by a corresponding degradation in signal quality - a common obstacle that is shared by most wearable technologies. We aim to resolve this issue by introducing a Deep Matched Filter (Deep-MF) for the highly accurate detection of R-peaks in wearable ECG, thus enhancing the utility of Ear-ECG in real-world scenarios. The Deep-MF consists of an encoder stage (trained as part of an encoder-decoder module to reproduce ground truth ECG), and an R-peak classifier stage. Through its operation as a Matched Filter, the encoder searches for matches with an ECG template pattern in the input signal, prior to filtering the matches with the subsequent convolutional layers and selecting peaks corresponding to true ECG matches. The so condensed latent representation of R-peak information is then fed into a simple R-peak classifier, of which the output provides precise R-peak locations. The proposed Deep Matched Filter is evaluated using leave-one-subject-out cross validation over 36 subjects with an age range of 18-75, with the Deep-MF outperforming existing algorithms for R-peak detection in noisy ECG. The Deep-MF achieves a median R-peak recall of 94.9\%, a median precision of 91.2\% and an (AUC) value of 0.97. Furthermore, we demonstrate that the Deep Matched Filter algorithm not only retains the initialised ECG kernel structure during the training process, but also amplifies portions of the ECG which it deems most valuable. Overall, the Deep Matched Filter serves as a valuable step forward for the real-world functionality of Ear-ECG and, through its explainable operation, the acceptance of deep learning models in e-health.

Amplitude-Independent Machine Learning for PPG through Visibility Graphs and Transfer Learning

Photoplethysmography (PPG) signals are omnipresent in wearable devices, as they measure blood volume variations using LED technology. These signals provide insight into the body's circulatory system and can be employed to extract various bio-features, such as heart rate and vascular ageing. Although several algorithms have been proposed for this purpose, many exhibit limitations, including heavy reliance on human calibration, high signal quality requirements, and a lack of generalization. In this paper, we introduce a PPG signal processing framework that integrates graph theory and computer vision algorithms, which is invariant to affine transformations, offers rapid computation speed, and exhibits robust generalization across tasks and datasets.

Convex Quaternion Optimization for Signal Processing: Theory and Applications

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 optimization. To this end, we establish an essential theory of convex quaternion optimization for signal processing based on the generalized Hamilton-real (GHR) calculus. This is achieved in a way which conforms with traditional complex and real optimization theory. For rigorous, We present five discriminant theorems for convex quaternion functions, and four discriminant criteria for strongly convex quaternion functions. Furthermore, we provide a fundamental theorem for the optimality of convex quaternion optimization problems, and demonstrate its utility through three applications in quaternion signal processing. These results provide a solid theoretical foundation for convex quaternion optimization and open avenues for further developments in signal processing applications.

UAdam: Unified Adam-Type Algorithmic Framework for Non-Convex Stochastic Optimization

Adam-type algorithms have become a preferred choice for optimisation in the deep learning setting, however, despite success, their convergence is still not well understood. To this end, we introduce a unified framework for Adam-type algorithms (called UAdam). This is equipped with a general form of the second-order moment, which makes it possible to include Adam and its variants as special cases, such as NAdam, AMSGrad, AdaBound, AdaFom, and Adan. This is supported by a rigorous convergence analysis of UAdam in the non-convex stochastic setting, showing that UAdam converges to the neighborhood of stationary points with the rate of $\mathcal{O}(1/T)$. Furthermore, the size of neighborhood decreases as $\beta$ increases. Importantly, our analysis only requires the first-order momentum factor to be close enough to 1, without any restrictions on the second-order momentum factor. Theoretical results also show that vanilla Adam can converge by selecting appropriate hyperparameters, which provides a theoretical guarantee for the analysis, applications, and further developments of the whole class of Adam-type algorithms.

Graph Tensor Networks: An Intuitive Framework for Designing Large-Scale Neural Learning Systems on Multiple Domains

Despite the omnipresence of tensors and tensor operations in modern deep learning, the use of tensor mathematics to formally design and describe neural networks is still under-explored within the deep learning community. To this end, we introduce the Graph Tensor Network (GTN) framework, an intuitive yet rigorous graphical framework for systematically designing and implementing large-scale neural learning systems on both regular and irregular domains. The proposed framework is shown to be general enough to include many popular architectures as special cases, and flexible enough to handle data on any and many data domains. The power and flexibility of the proposed framework is demonstrated through real-data experiments, resulting in improved performance at a drastically lower complexity costs, by virtue of tensor algebra.

Hearables: Ear EEG Based Driver Fatigue Detection

Ear EEG based driver fatigue monitoring systems have the potential to provide a seamless, efficient, and feasibly deployable alternative to existing scalp EEG based systems, which are often cumbersome and impractical. However, the feasibility of detecting the relevant delta, theta, alpha, and beta band EEG activity through the ear EEG is yet to be investigated. Through measurements of scalp and ear EEG on ten subjects during a simulated, monotonous driving experiment, this study provides statistical analysis of characteristic ear EEG changes that are associated with the transition from alert to mentally fatigued states, and subsequent testing of a machine learning based automatic fatigue detection model. Novel numerical evidence is provided to support the feasibility of detection of mental fatigue with ear EEG that is in agreement with widely reported scalp EEG findings. This study paves the way for the development of ultra-wearable and readily deployable hearables based driver fatigue monitoring systems.