Phase retrieval in quaternion domains presents unique computational challenges due to the non-commutative nature of quaternion algebra. Although several methods have been developed to address this issue, there remains significant potential for improvement. In this paper, we develop a novel quaternion-based alternating direction method of multipliers framework for quaternion phase retrieval from magnitude-only measurements based on the generalized Hamilton-real calculus, and provide a rigorous convergence analysis. Through extensive numerical simulations, we demonstrate that the proposed quaternion-based alternating direction method of multipliers framework outperforms existing methods, such as quaternion Wirtinger flow, quaternion truncated Wirtinger flow, and quaternion truncated amplitude flow, thereby establishing a new benchmark for quaternion phase retrieval problems. The theoretical framework and empirical results demonstrate that quaternion-based alternating direction method of multipliers offers a novel and effective approach for solving quaternion phase retrieval problems.

Nuclear receptors (NRs) are a class of essential proteins that regulate the expression of specific genes and are associated with multiple diseases. In silico methods for prescreening potential NR binders with predictive binding ability are highly desired for NR-related drug development but are rarely reported. Here, we present the PbsNRs (Predicting binders and scaffolds for Nuclear Receptors), a user-friendly web server designed to predict the potential NR binders and scaffolds through proteochemometric modeling. The utility of PbsNRs was systemically evaluated using both chemical compounds and natural products. Results indicated that PbsNRs achieved a good prediction performance for chemical compounds on internal (ROC-AUC = 0.906, where ROC is Receiver-Operating Characteristic curve and AUC is the Area Under the Curve) and external (ROC-AUC = 0.783) datasets, outperforming both compound-ligand interaction tools and NR-specific predictors. PbsNRs also successfully identified bioactive chemical scaffolds for NRs by screening massive natural products. Moreover, the predicted bioactive and inactive natural products for NR2B1 were experimentally validated using biosensors. PbsNRs not only aids in screening potential therapeutic NR binders but also helps discover the essential molecular scaffold and guide the drug discovery for multiple NR-related diseases. The PbsNRs web server is available at http://pbsnrs.badd-cao.net.

Shuffling gradient algorithms are extensively used to solve finite-sum optimization problems in machine learning. However, their theoretical properties still need to be further explored, especially the last-iterate convergence in the non-convex setting. In this paper, we study the last-iterate convergence behavior of shuffling momentum gradient (SMG) method, a shuffling gradient algorithm with momentum. Specifically, we focus on the non-convex scenario and provide theoretical guarantees under arbitrary shuffling strategies. For non-convex objectives, we achieve the convergence of gradient norms at the last-iterate, showing that every accumulation point of the iterative sequence is a stationary point of the non-convex problem. Our analysis also reveals that the function values of the last-iterate converge to a finite value. Additionally, we obtain the asymptotic convergence rates of gradient norms at the minimum-iterate. By employing a uniform without-replacement sampling strategy, we further achieve an improved convergence rate for the minimum-iterate output. Under the Kurdyka-Lojasiewicz (KL) inequality, we establish the challenging strong limit-point convergence results. In particular, we prove that the whole sequence of iterates exhibits convergence to a stationary point of the finite-sum problem. By choosing an appropriate stepsize, we also obtain the corresponding rate of last-iterate convergence, matching available results in the strongly convex setting. Given that the last iteration is typically preferred as the output of the algorithm in applied scenarios, this paper contributes to narrowing the gap between theory and practice.

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.

Adam-type algorithms have become a preferred choice for optimization in the deep learning setting; however, despite their success, their convergence is still not well understood. To this end, we introduce a unified framework for Adam-type algorithms, termed UAdam. It is equipped with a general form of the second-order moment, which makes it possible to include Adam and its existing and future variants as special cases, such as NAdam, AMSGrad, AdaBound, AdaFom, and Adan. The approach is supported by a rigorous convergence analysis of UAdam in the general nonconvex stochastic setting, showing that UAdam converges to the neighborhood of stationary points with a rate of O(1/T). Furthermore, the size of the neighborhood decreases as the parameter β1 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 reveal the convergence conditions of vanilla Adam, together with the selection of appropriate hyperparameters. This provides a theoretical guarantee for the analysis, applications, and further developments of the whole general class of Adam-type algorithms. Finally, several numerical experiments are provided to support our theoretical findings.

This letter proposes a quaternion projection gradient ascent (QPGA) iterative algorithm based on generalized$\mathbb {HR}$calculus for computing the principal eigenvalues and its eigenvectors of quaternion Hermitian matrices. We also prove the convergence of the QPGA algorithm, demonstrating that the estimated sequence of principal eigenvalues is monotonically increasing. Numerical experiments demonstrate the superiority of the proposed iterative method over traditional algebraic methods in terms of accuracy and speed, as well as the application of principal eigenvalues and their eigenvectors obtained by the QPGA algorithm in denoising with quaternion principal component analysis and quaternion least mean square (QLMS) algorithms in filtering fetal electrocardiograms. Overall, the fast quaternion eigenvalue solving method provides a novel and effective technical tool for quaternion signal processing.

Price's theorem in statistical signal processing relates the expectation of a nonlinear function of normally distributed random variables to their covariances. However, such a key theorem is yet to be established and explored in quaternion statistics. To this end, we introduce Price's theorem for quaternion variables using the generalized Hamilton-real (GHR) calculus. This is achieved by first employing the chain rule of GHR calculus to derive two crucial quaternion matrix derivatives of functions with respect to the product of quaternion matrix variables. Next, we leverage quaternion second-order statistics to establish the relationship between the derivative of a function with respect to the augmented quaternion covariance matrix and its real counterpart. Based on the above results and Price's theorem in real signal processing, we finally propose a novel formulation of Price's theorem for quaternion random variables. This finding not only enriches the theory of quaternion statistical signals processing but also extends its applicability.

In this article, we investigate the boundedness and convergence of the online gradient method with the smoothing group regularization for the sigma-pi-sigma neural network (SPSNN). This enhances the sparseness of the network and improves its generalization ability. For the original group regularization, the error function is nonconvex and nonsmooth, which can cause oscillation of the error function. To ameliorate this drawback, we propose a simple and effective smoothing technique, which can effectively eliminate the deficiency of the original group regularization. The group regularization effectively optimizes the network structure from two aspects redundant hidden nodes tending to zero and redundant weights of surviving hidden nodes in the network tending to zero. This article shows the strong and weak convergence results for the proposed method and proves the boundedness of weights. Experiment results clearly demonstrate the capability of the proposed method and the effectiveness of redundancy control. The simulation results are observed to support the theoretical results.

Widely linear modeling is an important quaternion signal processing technique for capturing the complete second-order statistics of quaternion signals. However, the algorithms based on widely linear modeling are computationally expensive due to the augmented variables and statistics. In this letter, a fast estimation technique based on a real augmented representation of widely linear modeling is proposed in the context of quaternion extreme learning machine with augmented hidden layer (QELMAH), resulting in a reduction of almost 93% of multiplications and 75% of additions. An equivalence between the proposed algorithm and the original QELMAH is theoretically established by proving that the trained networks with the proposed algorithm and the original one mathematically implement identical mapping. Such a technique is also applicable to the widely linear quaternion recursive least squares algorithm. The theoretical analysis and the effectiveness of the proposed algorithms are validated by simulations on two benchmark problems.

Foot-and-mouth disease virus (FMDV) is an antigenic-variable RNA virus that is responsible for the recurrence of foot-and-mouth disease in livestock and can be prevented and controlled using a vaccine with broad-spectrum protection. Current anti-genicity evaluation methods, which involve animal immunity experiments and serum preparation, are unable to fulfill the needs of high-throughput antigenicity measurements. This study designed an antigenicity scoring model to rapidly predict the antigenicity of FMDV. Antigenic-dominant sites were initially determined on the VP1 protein, a position-specific scoring matrix and physical chemical indexes were integrated to generate antigenicity descriptors. Independent tests showed a high accuracy of 0.848 and an AUC value of 0.889, indicating the good performance of the model in antigenicity measurement. When applying this model to historical data, annual antigenicity coverage of widely used vaccine strains was successfully evaluated, this was also supported by previous experiments. Furthermore, the utility of this model was extended to select potential broad-spectrum vaccines among 1,201 historical non-redundant strains to recommend potential univalent, bivalent and trivalent vaccine candidates. The results suggested that the computational model designed in this study could be used for the high-throughput antigenicity measurement of FMDV and could aid in vaccine development for preventing FMDV epidemics.

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks.

A Quaternion Least Mean Phase (QLMP) algorithm is introduced for phase-only adaptive filtering of quaternion-valued signals. This is achieved by first defining the notion of phase in the quaternion domain based on the exponential representation of quaternion random variables, and then using the HR-calculus to derive a steepest-decent weight update. The advantages of this adaptive algorithm are illustrated in a bearings only tracking scenario, where the QLMP is shown to outperform the amplitude-phase based quaternion Least Mean Square (QLMS).

We introduce a novel and unifying framework for the calculation of gradients of both quaternion holomorphic functions and nonholomorphic real functions of quaternion variables. This is achieved by considering the isomorphism between the quaternion domain H and the bivariate complex domain C×C, and by exploiting complex calculus to simplify the quaternion gradient calculation. The validation of the proposed HC calculus is performed against the existing HR calculus, and its convenience is illustrated in the context of gradient-based quaternion optimisation as well as in adaptive learning systems. Quaternion adaptive filtering algorithms and a dynamical perceptron update are next derived based on the bivariate complex representation of quaternions and the HC calculus. Simulations on both synthetic and real-world multidimensional signals support the analysis.

The multi-valued neuron (MVN) has a strong multi-classification ability. However, the MVN learning algorithms require the complex-valued learning rate and depends on the unknown optimal weights. To address this issue, we introduce a modified MVN that centers the neuron state in each sector. The learning algorithms of the modified MVN are able to reuse the real-valued learning rate and eliminate the dependencies on the optimal weights. We prove the convergence of the modified MVN learning algorithms with real-valued learning rate.

This letter presents a unified convergence analysis of the split-complex nonlinear gradient descent (SCNGD) learning algorithms for complex-valued recurrent neural networks, covering three classes of SCNGD algorithms: standard SCNGD, normalized SCNGD, and adaptive normalized SCNGD. We prove that if the activation functions are of split-complex type and some conditions are satisfied, the error function is monotonically decreasing during the training iteration process, and the gradients of the error function with respect to the real and imaginary parts of the weights converge to zero. A strong convergence result is also obtained under the assumption that the error function has only a finite number of stationary points. The simulation results are given to support the theoretical analysis.