Real-valued Matrix Research Articles

I/O-efficient iterative matrix inversion with photonic integrated circuits

Photonic integrated circuits have been extensively explored for optical processing with the aim of breaking the speed and energy efficiency bottlenecks of digital electronics. However, the input/output (IO) bottleneck remains one of the key barriers. Here we report a photonic iterative processor (PIP) for matrix-inversion-intensive applications. The direct reuse of inputted data in the optical domain unlocks the potential to break the IO bottleneck. We demonstrate notable IO advantages with a lossless PIP for real-valued matrix inversion and integral-differential equation solving, as well as a coherent PIP with optical loops integrated on-chip, enabling complex-valued computation and a net inversion time of 1.2 ns. Furthermore, we estimate at least an order of magnitude enhancement in IO efficiency of a PIP over photonic single-pass processors and the state-of-the-art electronic processors for reservoir training tasks and multiple-input and multiple-output (MIMO) precoding tasks, indicating the huge potential of PIP technology in practical applications.

On-chip multifunctional self-configurable quadrilateral MZI network

Photonic integrated circuits have garnered significant attention in recent years. To enhance the functional versatility of these devices, researchers have introduced the concept of reconfiguration into photonic integrated circuits. Inspired by field programmable gate arrays in the electrical domain, programmable photonic chips employing various topologies have been developed. However, users still encounter challenges when utilizing these devices, as they need to understand the internal structure and principles of the chip and individually adjust the tunable basic units within the topology network. In this paper, we employ the quadrilateral topological network based on the on-chip Mach–Zehnder interferometer as a black box to realize a highly self-reconfigurable optical signal processor. By leveraging this approach, we achieve positive real-valued matrix computation, optical routing, and low-loss optical energy splitting. Our demonstration effectively showcases the immense potential of on-chip programmable photonic waveguide meshes.

Spectrally Efficient Non-Orthogonal Discrete Multi-Tone Transmission for Underwater Wireless Optical Communication With Low-Complexity High Performance ICI Mitigation

In this paper, a novel spectrally efficient non-orthogonal discrete multi-tone (NODMT) system is proposed for the bandwidth-starved underwater wireless optical communication (UWOC) to compensate for the insufficient bandwidth of components. Different from other non-orthogonal systems with complex-valued correlation matrices, the designed NODMT system has a real-valued correlation matrix, which is significant in reducing the complexity of decoders, e.g., the complexity of the proposed NODMT with iterative detection (ID) algorithm can be reduced by 50% as compared to the conventional non-orthogonal system with ID. However, in circumstances of higher spectrum efficiency (SE), the non-orthogonality is intensified and inter-carrier interference (ICI) becomes severer, leading to the sharp deterioration of ID decoding capability. The traditional sphere decoding (SD) algorithm cannot solve this issue because Cholesky decomposition only works with the positive definite matrix. Therefore, we propose a novel SD algorithm based on QR decomposition, named QRSD, which effectively compensates for the insufficient decoding capability of the ID algorithm in the case of higher SE. By combining ID and QRSD methods, over 40% complexity reduction and nearly identical BER performance with QRSD are achieved. Through a proof-of-concept UWOC experiment, a 3.31-Gbps NODMT transmission with up to 24.44% data rate promotion is demonstrated, and the feasibility / effectiveness of the proposed ID-QRSD algorithm in different turbidity water is verified. Moreover, we have experimentally realized 75-m/2.66-Gbps and 60-m/3.02-Gbps underwater transmission with 15% and 25% bandwidth savings, respectively. This UWOC system with high SE shows great potential in the bandwidth-limited UWOC and underwater internet of things (UIoT).

Highly Efficient Numerical Algorithm for Nonlinear Space Variable-Order Fractional Reaction–Diffusion Models

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability.

A Novel Complex-Valued Gaussian Measurement Matrix for Image Compressed Sensing.

The measurement matrix used influences the performance of image reconstruction in compressed sensing. To enhance the performance of image reconstruction in compressed sensing, two different Gaussian random matrices were orthogonalized via Gram-Schmidt orthogonalization, respectively. Then, one was used as the real part and the other as the imaginary part to construct a complex-valued Gaussian matrix. Furthermore, we sparsified the proposed measurement matrix to reduce the storage space and computation. The experimental results show that the complex-valued Gaussian matrix after orthogonalization has better image reconstruction performance, and the peak signal-to-noise ratio and structural similarity under different compression ratios are better than the real-valued measurement matrix. Moreover, the sparse measurement matrix can effectively reduce the amount of calculation.

Quaternion Factorization Machines: A Lightweight Solution to Intricate Feature Interaction Modeling.

Due to the sparsity of available features in web-scale predictive analytics, combinatorial features become a crucial means for deriving accurate predictions. As a well-established approach, a factorization machine (FM) is capable of automatically learning high-order interactions among features to make predictions without the need for manual feature engineering. With the prominent development of deep neural networks (DNNs), there is a recent and ongoing trend of enhancing the expressiveness of FM-based models with DNNs. However, though better results are obtained with DNN-based FM variants, such performance gain is paid off by an enormous amount (usually millions) of excessive model parameters on top of the plain FM. Consequently, the heavy parameterization impedes the real-life practicality of those deep models, especially efficient deployment on resource-constrained Internet of Things (IoT) and edge devices. In this article, we move beyond the traditional real space where most deep FM-based models are defined and seek solutions from quaternion representations within the hypercomplex space. Specifically, we propose the quaternion factorization machine (QFM) and quaternion neural factorization machine (QNFM), which are two novel lightweight and memory-efficient quaternion-valued models for sparse predictive analytics. By introducing a brand new take on FM-based models with the notion of quaternion algebra, our models not only enable expressive inter-component feature interactions but also significantly reduce the parameter size due to lower degrees of freedom in the hypercomplex Hamilton product compared with real-valued matrix multiplication. Extensive experimental results on three large-scale datasets demonstrate that QFM achieves 4.36% performance improvement over the plain FM without introducing any extra parameters, while QNFM outperforms all baselines with up to two magnitudes' parameter size reduction in comparison to state-of-the-art peer methods.

Simultaneous ruin probability for multivariate Gaussian risk model

Let Z(t)=(Z1(t),…,Zd(t))⊤,t∈R where Zi(t),t∈R, i=1,…,d are mutually independent centered Gaussian processes with continuous sample paths a.s. and stationary increments. For X(t)=AZ(t),t∈R, where A is a nonsingular d×d real-valued matrix, u,c∈Rd and T>0 we derive tight bounds for P∃t∈[0,T]:∩i=1d{Xi(t)−cit>ui}and find exact asymptotics as (u1,…,ud)⊤=(ua1,…,uad)⊤ for any (a1,…,ad)⊤∈Rd∖(−∞,0]d and u→∞.

Relativistic Fermion and Boson Fields: Bose-Einstein Condensate as a Time Crystal

In a basis of the space-time coordinate frame four quaternions discovered by Hamilton can be used. For subsequent reproduction of the coordinate frame these four quaternions are expanded to four 4 × 4 matrices with real-valued matrix coefficients −0 and 1. This group set is isomorphic to the SU(2) group. Such a matrix basis introduces extra six degrees of freedom of matter motion in space-time. There are three rotations about three space axes and three boosts along these axes. Next one declares the differential generating operators acting on the energy-momentum density tensor written in the above quaternion basis. The subsequent actions of this operator together with its transposed one on the above tensor lead to the emergence of the gravitomagnetic equations that are like the Maxwell equations. Wave equations extracted from the gravitomagnetic ones describe the propagation of energy density waves and their vortices through space. The Dirac equations and their reduction to two equations with real-valued functions, the quantum Hamilton-Jacobi equations and the continuity equations, are considered. The Klein-Gordon equations arising on the mass shell hints to the alternation of the paired fermion fields and boson ones. As an example, a Feynman diagram of an electron–positron time crystal is illustrated.

System decomposition-based stability criteria for Takagi-Sugeno fuzzy uncertain stochastic delayed neural networks in quaternion field

<abstract><p>Stochastic disturbances often occur in real-world systems which can lead to undesirable system dynamics. Therefore, it is necessary to investigate stochastic disturbances in neural network modeling. As such, this paper examines the stability problem for Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks. By applying Takagi-Sugeno fuzzy models and stochastic analysis, we first consider a general form of Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks with time-varying delays. Then, by constructing suitable Lyapunov-Krasovskii functional, we present new delay-dependent robust and global asymptotic stability criteria for the considered networks. Furthermore, we present our results in terms of real-valued linear matrix inequalities that can be solved in MATLAB LMI toolbox. Finally, two numerical examples are presented with their simulations to demonstrate the validity of the theoretical analysis.</p></abstract>

Real-Valued Optical Matrix Computing with Simplified MZI Mesh

The Mach–Zehnder interferometer (MZI) mesh, a mainstream structure for optical matrix-vector multiplication (MVM), has been widely employed in recently developed optical neural networks (ONNs) and combination optimization problem solvers. The conventional MZI mesh was designed specifically for complex-valued optical MVM. The network includes 2 N 2 phase shifters, and coherent detection is indispensable for retrieving the output complex-valued vectors. Nonetheless, the majority of applications, including ONNs, merely require real-valued optical matrices with N 2 degrees of freedom (DOFs). The DOF gap between the 2 types of matrices results in a severe redundancy in the number of phase shifters when the conventional MZI mesh is applied to implement real-valued optical MVM. In this study, we propose a simplified MZI mesh for performing real-valued incoherent optical MVM. It has N 2 phase shifters and an optical depth of N + 1, and it outperforms the conventional MZI mesh. Furthermore, we constructed an ONN with the proposed MZI mesh and successfully performed the iris classification task via in situ training of particle swarm optimization. More importantly, we introduced a matched on-chip nonlinear activation function, so the proposed MZI mesh can be cascaded onto a single chip. Overall, the proposed real-valued MZI mesh and in situ training method are space efficient, energy efficient, scalable, and robust to fabrication errors. Therefore, they are suitable for large-scale ONNs.

Dynamic Synchrophasor Estimation Based on Weighted Real-Valued Sinc Interpolation Method

Phasor measurement units (PMUs) are the cornerstone of intelligent sensing and monitoring of power systems. The superiority of the sinc interpolation function-based least squares (SLS) method, a signal processing algorithm for PMUs, has been proven in harmonic phasor estimation. However, the SLS-based estimators suffer from spectral interferences and heavy computational burden. To this end, a real-valued sinc weighted least squares (SWLS-R) method is proposed to track the dynamic synchrophasor accurately and quickly. First, the weighted vector, i.e., window function, is introduced to enhance the SLS estimator’s behavior by mitigating the impact of spectral interference. Then, the complex-valued system matrix is transformed into the real-valued system matrix to lower the computational burden by avoiding complex matrix multiplication. Finally, an explicit and fast solution is designed to calculate the pseudoinverse matrix of the SWLS-R, which aims to further improve computational efficiency. The accuracy and effectiveness of the proposed method are validated by simulations. These are designed, under steady and dynamic conditions, according to the M-class requirements of the Standard IEC/IEEE 60255-118-1.

System decomposition method-based global stability criteria for T-S fuzzy Clifford-valued delayed neural networks with impulses and leakage term

<abstract><p>This paper investigates the global asymptotic stability problem for a class of Takagi-Sugeno fuzzy Clifford-valued delayed neural networks with impulsive effects and leakage delays using the system decomposition method. By applying Takagi-Sugeno fuzzy theory, we first consider a general form of Takagi-Sugeno fuzzy Clifford-valued delayed neural networks. Then, we decompose the considered $ n $-dimensional Clifford-valued systems into $ 2^mn $-dimensional real-valued systems in order to avoid the inconvenience caused by the non-commutativity of the multiplication of Clifford numbers. By using Lyapunov-Krasovskii functionals and integral inequalities, we derive new sufficient criteria to guarantee the global asymptotic stability for the considered neural networks. Further, the results of this paper are presented in terms of real-valued linear matrix inequalities, which can be directly solved using the MATLAB LMI toolbox. Finally, a numerical example is provided with their simulations to demonstrate the validity of the theoretical analysis.</p></abstract>

Sample-level weights learning for multi-view clustering on spectral rotation

Multi-view clustering usually yields better results than single-view clustering since it utilizes more information from multi-view data. However, in the original multi-view data, some samples may be corrupted in partial views. Under this situation, the locations of the corrupted data are often unknown. But there is limited literature regarding this problem. Moreover, most existing multi-view spectral clustering methods need a post-processed algorithm k-means after obtaining the partition matrix, which leads to deviations in the clustering results. To resolve these problems, we propose a multi-view spectral clustering method named sample-level weights learning for Multi-view Clustering on Spectral Rotation (SR-MC) in this paper. By learning the weights in the sample level, SR-MC can make full use of the helpful complementary information among different views while reducing the effects of low-quality data for each sample. Therefore, it can deal with various multi-view clustering scenarios such as data are complete or corrupted in partial views. To reduce the deviations of the clustering results, a joint framework is designed for combining the learning of the consensus Laplacian matrix, the real-valued partition matrix and the binary indicator matrix together. The objective function of SR-MC can be efficiently optimized by an alternative optimization algorithm. Compared with other ten baselines, experiments on seven datasets show the superiority of our method.

System Monotonicity and Subspace Tracking: A Geometric Perspective of the Frisch–Shapiro Scheme

The Shapiro scheme, together with the closely related Frisch–Kalman scheme, has been an important approach to system identification and statistical analysis. A longstanding result on this scheme, known as the Shapiro theorem, is both informative and significant. This article imparts a geometric understanding to the Shapiro theorem and generalizes it to the asymmetric setting using the notion of cone-invariance. In particular, we establish the equivalence between two important properties of a real-valued square matrix—irreducible orthant-invariance and simplicity of its dominant eigenvalue under arbitrary diagonal perturbations. The result can be regarded as a converse Perron–Frobenius theorem. Furthermore, we investigate two applications of the proposed result in systems and control, namely, characterization of irreducibly orthant-monotone nonlinear systems and subspace tracking via decentralized control. We also extend the established result to accommodating polyhedral cones and obtain several insights.

A Coreless Current Probe for Multicore Cables

To measure the current on each conductor in a three-phase three-core cable or a three-phase four-core cable without an armor or the armor that is made of nonmagnetic material, one has to remove its sheath/armor and applies a traditional device, such as a current transformer (CT) to each conductor, which is invasive and inconvenient. In this article, a Coreless current probe, composed of a tunnel-magnetoresistance (TMR)-based magnetic sensor array, is presented to nondestructively achieve current measurement for multicore cables. The conductor positions, incline angles, and current phasors of a multicore cable are simultaneously identified by solving a nonlinear least square (NLLS) problem. The current waveforms are reconstructed in real time through a real-valued matrix multiplication using the measured time-domain TMR sensor voltage signals. Both simulation and experiment are performed to validate the feasibility of the proposed method. The simulation study is first conducted and demonstrates that a maximum relative current amplitude error of less than 5% and a maximum phase error of lower than 4° can be achieved, and three-phase current waveforms with many harmonics are also successfully reconstructed by simulation. A prototype Coreless current probe is designed and fabricated. The experimental study involving fundamental current phasor measurement and current waveform reconstruction is subsequently performed on a three-phase three-core cable. The experimental results indicate that the maximum relative current amplitude error and phase error, respectively, have average values of 5.44% and 2.58°. Meanwhile, three-phase current waveforms with many harmonics generated by three types of appliances are also successfully reconstructed.

A low‐computational‐cost alternating switches algorithm for real‐valued spatial covariance matrix reconstruction

The application of the emerging hybrid analog–digital architecture for future millimeter-wave communications has attracted significant attention. Although this architecture can reduce the hardware cost and power consumption considerably, the high-resolution direction of arrival (DOA) estimation in the architecture based on switches is still a challenge. As a solution, an alternating switches algorithm (ASA) is proposed in this paper. By appropriately adjusting the switch connected to each antenna, the real-valued spatial covariance matrix (SCM) is reconstructed with low-dimensional vector-to-vector multiplication. Subsequently, the DOA angle of each signal is estimated based on a split subspace extracted from the real-valued SCM. Finally, a set of simulation experiments are performed to verify the performance of the proposed algorithm. The results indicate that the ASA can reconstruct the real-valued SCM and realize high-resolution DOA estimation with a substantially low computational cost.

Splitting games over finite sets

This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales \(\{p_t,q_t\}_t\), in order to control a terminal payoff \(u(p_\infty ,q_\infty )\). A first part introduces the notion of “Mertens–Zamir transform" of a real-valued matrix and use it to approximate the solution of the Mertens–Zamir system for continuous functions on the square \([0,1]^2\). A second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state: building on Laraki and Renault (Math Oper Res 45:1237–1257, 2020), we show that the value exists by constructing non Markovian \(\varepsilon \)-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.

A Ternary Neural Network with Compressed Quantized Weight Matrix for Low Power Embedded Systems

In this paper, we propose a method of transforming a real-valued matrix to a ternary matrix with controllable sparsity. The sparsity of quantized weight matrices can be controlled by adjusting the threshold during the training and quantizing process. A 3-layer ternary neural network was trained with the MNIST dataset using the proposed adjustable dynamic threshold. The sparsity of the quantized weight matrices varied from 0.1 to 0.6 and the obtained recognition rate reduced from 91% to 88%. The sparse weight matrices were compressed by the compressed sparse row format to speed up the ternary neural network, which can be deployed on low-power embedded systems, such as the Raspberry Pi 3 board. The ternary neural network with the sparsity of quantized weight matrices of 0.1 is 4.24 times faster than the ternary neural network without compressing weight matrices. The ternary neural network is faster as the sparsity of quantized weight matrices increases. When the sparsity of the quantized weight matrices is as high as 0.6, the recognition rate degrades by 3%, however, the speed is 9.35 times the ternary neural network's without compressing quantized weight matrices. Ternary neural network work with compressed sparse matrices is feasible for low-cost, low-power embedded systems.

Polarization-Aware Prediction of Mobile Radio Wave Propagation Based on Complex-Valued and Quaternion Neural Networks

It is important to predict the characteristics of radio waves propagating in wireless environment for compensating information loss caused by fading effects. Though fading is often analyzed as the changes of intrinsic electric-field phasor describing a communication channel, it is also attributed greatly to its polarization state (PS). In this paper, we propose PS and phasor prediction methods based on quaternion neural networks (QNNs) and complex-valued neural networks (CVNNs). Quaternion is a hypercomplex number system for representing three dimensional (3-D) spatial rotation and orientation. It realizes adaptive prediction more stably and effectively than a real-valued rotation matrix. The time trajectories of PS are expressed as 3-D rotational motions on the Poincare sphere. We show that QNNs learn the spatiotemporal movements of the PS with high generalization ability and thus provide a superior prediction performance. In addition, CVNNs are efficient predictors for processing complex-valued signals such as the phasor of an incident wave. We evaluate the precision of the proposed QNN-CVNN prediction scheme in channel equalization in a time-division-duplex (TDD) orthogonal-frequency-division-multiplexing (OFDM) communication system. In physical wireless-environment experiments, we find that the proposed scheme provides 3.0 dB and 6.0 dB improvements of bit error rate (BER) at 10^-4, showing almost the same BER performance as a system perfectly knowing the actual radio-wave characteristics in future TDD frames. Accordingly, this work also reveals that PS needs to be considered as one of the significant characteristics in various radio-wave systems in the near-future technology.

Video-rate dual-modal photoacoustic and fluorescence imaging through a multimode fibre towards forward-viewing endomicroscopy

Multimode fibres (MMFs) are becoming increasingly attractive in optical endoscopy as they promise to enable unparallelled miniaturisation, spatial resolution and cost. However, high-speed imaging with wavefront shaping has been challenging. Here, we report the development of a video-rate dual-modal photoacoustic (PA) and fluorescence microscopy probe with a high-speed digital micromirror device (DMD) towards forward-viewing endomicroscopy. Optimal DMD patterns were obtained using a real-valued intensity transmission matrix algorithm to raster-scan a 1.5 μm-diameter focused beam at the distal fibre tip for imaging. The PA imaging speed and spatial resolution were varied from ∼2 to 57 frames per second and from 1.7 to 3 μm, respectively. Further, high-fidelity PA images of carbon fibres and mouse red blood cells were acquired at unprecedented speed. The capability of dual-modal imaging was demonstrated with phantoms. We anticipate that with further miniaturisation of the ultrasound detector, this probe could be integrated into medical needles to guide minimally invasive procedures.