Subspace Matrix Research Articles

SCAE: Structural Contrastive Auto-Encoder for Incomplete Multi-View Representation Learning

Describing an object from multiple perspectives often leads to incomplete data representation. Consequently, learning consistent representations for missing data from multiple views has emerged as a key focus in the realm of Incomplete Multi-view Representation Learning (IMRL). In recent years, various strategies, such as subspace learning, matrix decomposition, and deep learning, have been harnessed to develop numerous IMRL methods. In this article, our primary research revolves around IMRL, with a particular emphasis on addressing two main challenges. Firstly, we delve into the effective integration of intra-view similarity and contextual structure into a unified framework. Secondly, we explore the effective facilitation of information exchange and fusion across multiple views. To tackle these issues, we propose a deep learning approach known as Structural Contrastive Auto-Encoder (SCAE) to solve the challenges of IMRL. SCAE comprises two major components: intra-view structural representation learning and inter-view contrastive representation learning. The former involves capturing intra-view similarity by minimizing the Dirichlet energy of the feature matrix, while also applying spatial dispersion regularization to capture intra-view contextual structure. The latter encourages maximizing the mutual information of inter-view representations, facilitating information exchange and fusion across views. Experimental results demonstrate the efficacy of our approach in significantly enhancing model accuracy and robustly addressing IMRL problems. The code is available at https://github.com/limengran98/SCAE .

Mode-wise principal subspace pursuit and matrix spiked covariance model

Abstract This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection. These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyse the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data.

Convergence analysis of state estimation for UAV formation under GPS‐denied environments

AbstractIn this article, we focus on the state estimation and formation control problem for a multi‐UAV system, where only parts of UAVs have GPS measurements, that is, the anchors, and the other UAVs measure their relative positions with anchors. With a second‐order discrete‐time model of each UAV in the formation, we analyze the convergence of state estimation (including its expectation and covariance) for commonly used consensus‐based formation control protocols, under the effect of control and measurement noise. More specifically, based on the discrete‐time stability analysis, we obtain the sufficient and necessary conditions for the expectation stability of the formation control. Using the fixed point theorem in positive symmetric matrix subspace and observability analysis, we rigorously prove that the covariance of estimation is bounded if and only if the anchor node set is globally reachable. Based on our analysis, the formation control parameters and system topology can be designed to satisfy sufficient conditions, such that the desired formation pattern can be achieved for the multi‐UAV dynamic system under GPS‐denied environments. Simulation results corroborate the effectiveness of our theoretical results.

One‐bit DOA estimation in non‐uniform noise with alternating minimization method

AbstractThis letter presents a method to optimize the estimation of direction of arrival (DOA) using the uniform linear array (ULA) with one‐bit signals in the presence of non‐uniform noise. With the Toeplitz properties and the rank characteristics of the signal subspace matrix formed by the ULA, the alternating minimization (AM) method is employed to optimize the estimation problem. To address this challenge further, we utilize the singular value thresholding (SVT) method and the approximate projection method. The angle deviations caused by non‐uniform noise can be effectively corrected and a significant improvement in the estimation performance can be achieved. The feasibility and effectiveness of the proposed method are demonstrated through simulation results.

Automatic Estimation of the Interference Subspace Dimension Threshold in the Subspace Projection Algorithms of Magnetoencephalography Based on Evoked State Data.

A class of algorithms based on subspace projection is widely used in the denoising of magnetoencephalography (MEG) signals. Setting the dimension of the interference (external) subspace matrix of these algorithms is the key to balancing the denoising effect and the degree of signal distortion. However, most current methods for estimating the dimension threshold rely on experience, such as observing the signal waveforms and spectrum, which may render the results too subjective and lacking in quantitative accuracy. Therefore, this study proposes a method to automatically estimate a suitable threshold. Time-frequency transformations are performed on the evoked state data to obtain the neural signal of interest and the noise signal in a specific time-frequency band, which are then used to construct the objective function describing the degree of noise suppression and signal distortion. The optimal value of the threshold in the selected range is obtained using the weighted-sum method. Our method was tested on two classical subspace projection algorithms using simulation and two sensory stimulation experiments. The thresholds estimated by the proposed method enabled the algorithms to achieve the best waveform recovery and source location error. Therefore, the threshold selected in this method enables subspace projection algorithms to achieve the best balance between noise removal and neural signal preservation in subsequent MEG analyses.

A Low-Complexity Direction-of-Arrival Estimation Algorithm for Noncircular Signals via Subspace Rotation Technique

In this paper, we investigate the problem of the heavy computational burden of the direction-of-arrival (DoA) estimation for the noncircular (NC) signals. A novel low-complexity direction-of-arrival estimation algorithm for NC signals via subspace rotation technique (SRT) is proposed. The proposed algorithm divides the noise subspace matrix along its row direction into two submatrices, and the SRT is performed to get a new reduced-dimension noise subspace. Then, utilizing the separation of variables and the orthogonality between the reduced-dimension noise subspace and the space spanned by the columns of the extended manifold matrix, a new one-dimensional spectral search function is derived to estimate DoAs. As the size of the block matrices of the noise subspace matrix has a great impact on the computational complexity of the spectral search, the optimal number of rows of the block matrices is determined. The proposed algorithm not only avoids the two-dimensional spectral search but also efficiently removes the redundancy computations in the one-dimensional spectral search. Theoretical analysis and simulation results show that the proposed algorithm can significantly improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation for the NC signals, especially in scenarios where large numbers of sensors are applied.

Time-Varying Channel Estimation Scheme for Uplink MU-MIMO in 6G Systems

Channel estimation is a challenging issue for millimeter wave (mmWave) and massive multiple-input multiple-output (MIMO) in the future sixth generation (6G) wireless systems, where the conventional estimation schemes may fail to track the fast varying channels, especially in high-speed mobile scenarios. In this paper, a novel tensor-based uplink channel estimation scheme is proposed for multi-user MIMO (MU-MIMO) systems over time-varying channels. In the proposed scheme, a low-overhead pilot transmission scheme is designed to track the varying channel. The received uplink signal at the base station (BS) is formulated as a third-order tensor which admits a CANDECOMP/PARAFAC (CP) model. The CP decomposition issue is then solved using blind matrix decomposition, in which the special structures of signals in the time dimension and the matrix subspace are utilized. By exploiting low-rank structure of the signal tensor, the channel parameters (angles of arrival/departure, path gains, and Doppler shifts) are estimated from the factor matrices. Moreover, the proposed scheme is theoretically analyzed and it is guaranteed with low pilot overhead. Simulation results verify that the proposed scheme can outperform the compressed sensing (CS) based scheme and the iteration-based scheme in terms of accuracy and stability. The uniqueness of the proposed scheme is also verified in our simulation.

Quantum Krylov subspace algorithms for ground- and excited-state energy estimation

Quantum Krylov subspace diagonalization (QKSD) algorithms provide a low-cost alternative to the conventional quantum phase estimation algorithm for estimating the ground and excited-state energies of a quantum many-body system. While QKSD algorithms typically rely on using the Hadamard test for estimating Krylov subspace matrix elements of the form, $\langle \phi_i|e^{-i\hat{H}\tau}|\phi_j \rangle$, the associated quantum circuits require an ancilla qubit with controlled multi-qubit gates that can be quite costly for near-term quantum hardware. In this work, we show that a wide class of Hamiltonians relevant to condensed matter physics and quantum chemistry contain symmetries that can be exploited to avoid the use of the Hadamard test. We propose a multi-fidelity estimation protocol that can be used to compute such quantities showing that our approach, when combined with efficient single-fidelity estimation protocols, provides a substantial reduction in circuit depth. In addition, we develop a unified theory of quantum Krylov subspace algorithms and present three new quantum-classical algorithms for the ground and excited-state energy estimation problems, where each new algorithm provides various advantages and disadvantages in terms of total number of calls to the quantum computer, gate depth, classical complexity, and stability of the generalized eigenvalue problem within the Krylov subspace.

A penalized method of alternating projections for weighted low-rank hankel matrix optimization

Weighted low-rank Hankel matrix optimization has long been used to reconstruct contaminated signal or forecast missing values for time series of a wide class. The Method of Alternating Projections (MAP) (i.e., alternatively projecting to a low-rank matrix manifold and the Hankel matrix subspace) is a leading method. Despite its wide use, MAP has long been criticized of lacking convergence and of ignoring the weights used to reflect importance of the observed data. The most of known results are in a local sense. In particular, the latest research shows that MAP may converge at a linear rate provided that the initial point is close enough to a true solution and a transversality condition is satisfied. In this paper, we propose a globalized variant of MAP through a penalty approach. The proposed method inherits the favourable local properties of MAP and has the same computational complexity. Moreover, it is capable of handling a general weight matrix, is globally convergent, and enjoys local linear convergence rate provided that the cutting off singular values are significantly smaller than the kept ones. Furthermore, the new method also applies to complex data. Extensive numerical experiments demonstrate the efficiency of the proposed method against several popular variants of MAP.

Data-Driven Predictive Control With Switched Subspace Matrices for an SCR System

Selective catalytic reduction (SCR) systems are distributed systems with strong time-varying parameter characteristics such that an accurate model for it is difficult to establish. Its control task simultaneously achieving high NO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>x</i></sub> conversion efficiency and low NH <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> slip is a typical multi-objective and multi-constraint problem, which is suitable to be solved in the framework of the model predictive control (MPC). However, how to find a data-driven identification method based on the dynamic characteristics of an SCR system and a corresponding MPC method for satisfying its emission requirements remain a formidable challenge. The sufficient identification for the traditional identification method with fixed subspace model requires an excessively high order subspace matrix, such that a degradation in real-time performance is caused and the generality of the method under non-identification conditions is limited. In this paper, utilizing the transient data of the SCR system under the WHTC cycle, a novel identification method for some lower order subspace matrices excited by the segmented data referring to the dynamic of the ammonia coverage ratio is established. A corresponding predictive controller with the switched subspace matrices according to working conditions is designed in order to further improve its real-time performance, generality and robustness. The simulation results show that under the identification condition the proposed predictive controller compared to the traditional method can improve the emissions of NO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>x</i></sub> and NH <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> , that under the non-identification condition the proposed predictive controller can also improve the emissions and its optimization effects have better robustness to uncertainties of the transient cycle, and that the proposed predictive controller saves an significant computation time.

A novel multi-site damage localization method based on near-field signal subspace fitting using uniform linear sensor array

Damage diagnose imaging methods based on guided waves have been widely developed to depict the position and area of damage in structural health monitoring (SHM). Among them, compact sensor array based imaging methods have been gradually applied with high accuracy, such as phased array imaging, spatial filter imaging, multiple signal classification imaging and so on. However, when the multi-site damage appears nearly, the performance of these methods usually degrades. On one hand, limited to the array aperture, traditional methods cannot distinguish adjacent damage sites. On another hand, some subspace decomposition based methods also fail, suffering from the correlation between scattered array signals from different damage sites. To realize localization of multi-site damage, a novel near-field signal subspace fitting based multi-site damage localization method is proposed in this paper. This method constructs the cost function, which describe the equivalence relation between the signal subspace and array manifold matrix. Then localization of multi-site damage could be realized through multidimensional search for the optimization of cost function. The proposed method is verified on the aluminum panel with 3 damage sites. Experimental results show that the proposed method can realize multi-site damage localization with the angle errors less than 4° and the distance errors less than 35 mm.

Fast Target Localization Method for FMCW MIMO Radar via VDSR Neural Network

The traditional frequency-modulated continuous wave (FMCW) multiple-input multiple-output (MIMO) radar two-dimensional (2D) super-resolution (SR) estimation algorithm for target localization has high computational complexity, which runs counter to the increasing demand for real-time radar imaging. In this paper, a fast joint direction-of-arrival (DOA) and range estimation framework for target localization is proposed; it utilizes a very deep super-resolution (VDSR) neural network (NN) framework to accelerate the imaging process while ensuring estimation accuracy. Firstly, we propose a fast low-resolution imaging algorithm based on the Nystrom method. The approximate signal subspace matrix is obtained from partial data, and low-resolution imaging is performed on a low-density grid. Then, the bicubic interpolation algorithm is used to expand the low-resolution image to the desired dimensions. Next, the deep SR network is used to obtain the high-resolution image, and the final joint DOA and range estimation is achieved based on the reconstructed image. Simulations and experiments were carried out to validate the computational efficiency and effectiveness of the proposed framework.

Model predictive control using subspace model identification

This paper addresses the problem of designing and implementing a data-driven model based model predictive controller (MPC). In particular, we consider the problem where a subspace identification approach is utilized to determine a state-space model, while applying first-principles based knowledge in the model identification (denoted as the constrained subspace model). The incorporation of the first-principles based constraints in the subspace matrix Patel et al. (2020) often leads to a feed-through matrix being present. Such a model then is the best representation of the system dynamics, but does not lend itself readily to existing linear MPC formulations where the feed-through matrix is assumed to be zero. Thus, an existing linear MPC formulation is adapted to handle the feed through matrix. The superior performance of this MPC design, which can utilize the constrained subspace model, over existing approaches is demonstrated using a two tank chemical stirred tank reactor process.

Unsupervised Multi-View Clustering by Squeezing Hybrid Knowledge From Cross View and Each View

Multi-view clustering methods have been a focus in recent years because of their superiority in clustering performance. However, typical traditional multi-view clustering algorithms still have shortcomings in some aspects, such as removal of redundant information, utilization of various views and fusion of multi-view features. In view of these problems, this paper proposes a new multi-view clustering method, low-rank subspace multi-view clustering based on adaptive graph regularization. We construct two new data matrix decomposition models into a unified optimization model. In this framework, we address the significance of the common knowledge shared by the cross view and the unique knowledge of each view by presenting new low-rank and sparse constraints on the sparse subspace matrix. To ensure that we achieve effective sparse representation and clustering performance on the original data matrix, adaptive graph regularization and unsupervised clustering constraints are also incorporated in the proposed model to preserve the internal structural features of the data. Finally, the proposed method is compared with several state-of-the-art algorithms. Experimental results for five widely used multi-view benchmarks show that our proposed algorithm surpasses other state-of-the-art methods by a clear margin.

A Tensor Subspace Representation-Based Method for Hyperspectral Image Denoising

In hyperspectral image (HSI) denoising, subspace-based denoising methods can reduce the computational complexity of the denoising algorithm. However, the existing matrix subspaces, which are generated by the unfolding matrix of the HSI tensor, cannot completely represent a tensor since the unfolding operation will destroy the tensor structure. To overcome this, we design a novel basis tensor that is directly learned from the original tensor and present a tensor subspace representation (TenSR), which is a more authentic representation for delivering the intrinsic structure of the tensor than a matrix subspace representation. Equipped with the TenSR, we then propose a TenSR-based HSI denoising (TenSRDe) model, which simultaneously considers the low-tubal rankness of the HSI tensor and the nonlocal self-similarity of the coefficient tensor. Moreover, we develop an efficient proximal alternating minimization (PAM) algorithm to solve the proposed nonconvex model and theoretically prove that the algorithm globally converges to a critical point. Experiments implemented on simulated and real data sets substantiate the denoising effect and efficiency of the proposed method.

Discriminative subspace matrix factorization for multiview data clustering

In a real-world scenario, an object is easily considered as features combined by multiple views in reality. Thus, multiview features can be encoded into a unified and discriminative framework to achieve satisfactory clustering performance. An increasing number of algorithms have been proposed for multiview data clustering. However, existing multiview methods have several drawbacks. First, most multiview algorithms focus only on origin data in high dimension directly without the intrinsic structure in the relative low-dimensional subspace. Spectral and manifold-based methods ignore pseudo-information that can be extracted from the optimization process. Thus, we design an unsupervised nonnegative matrix factorization (NMF)-based method called discriminative multiview subspace matrix factorization (DMSMF) for clustering. We provide the following contributions. (1) We extend linear discriminant analysis and NMF to a multiview version and connect them to a unified framework to learn in the discriminant subspace. (2) We propose a multiview manifold regularization term and discriminant multiview manifold regularization term that instruct the regularization term to discriminate different classes and obtain the geometry st ructure from the low-dimensional subspace. (3) We design an effective optimization algorithm with proven convergence to obtain an optimal solution procedure for the complex model. Adequate experiments are conducted on multiple benchmark datasets. Finally, we demonstrate that our model is superior to other comparable multiview data clustering algorithms.

A Cauchy score DOA estimator for monostatic MIMO radar in impulsive noise environment

ABSTRACT It is interesting to determine the direction of arrival (DOA) of signals in the presence of impulsive noise for monostatic multiple-input multiple-output (MIMO) radar. Inspired by the robust statistics property of M-estimation theory and maximum likelihood estimation theory, a Cauchy score cost function is proposed to suppress the outliers, leading to robust DOA estimation. Firstly, the Cauchy score cost function for residual fitting error matrix of the radar data is derived. Then, an alternating convex optimisation algorithm based on the complex-valued Newton gradient descent is tailored to acquire an efficient solution for the Cauchy score cost function. Finally, using the signal subspace matrix obtained by the alternating convex optimisation algorithm, ESPRIT algorithm is adopted to provide a closed-form DOA estimations. Simulation results are presented to demonstrate the effectiveness of the proposed method.

Generalised maximum complex correntropy‐based DOA estimation in presence of impulsive noise

Most existing methods for direction-of-arrival (DOA) estimation are severely influenced by impulsive noise due to their Gaussian noise assumption. As a typical non-linear similarity measure, the maximum correntropy criterion (MCC) has been considered to restrain impulsive noise, because of its ability to exploit the high-order statistics of signal. However, Gaussian kernel-based MCC method is not always the optimal choice and is only suitable for the real-valued signal, which certainly limits its applications. To solve the aforementioned problems, in this study, the authors proposed a novel generalised maximum complex correntropy criterion (GMCCC)-based complex-valued quasi-Newton method to restrain impulsive noise. GMCCC adopts the generalised complex Gaussian density function as the kernel function with more flexible parameters. Besides, it can extend the benefit to the complex-valued signal. Furthermore, its properties are formalised. The complex-valued quasi-Newton method guarantees the positive definite Hessian matrix to achieve the alternate minimisation of signal subspace and signal matrix. GMCCC achieves the accurate DOA estimation from the received data which does not require the covariance matrix. Stability performance and convergence are analysed. Experiment results show that the proposed GMCCC algorithm possesses the robustness and outperforms the state-of-the-art algorithms.

Low-Complexity Subspace MMSE Channel Estimation in Massive MU-MIMO System

Massive multi-user multiple-input multiple-output (massive MU-MIMO) technology is considered as a promising enabler to fulfill the rapid growth of traffic requirement for wireless mobile communications. The massive MU-MIMO system can achieve unlimited capacity when the base station (BS) has accurate channel state information (CSI). In time-division-duplex (TDD) mode, the BS estimates CSI by receiving pilot signals sent from user terminals (UEs). However, because of using non-orthogonal pilots, pilot contamination happens to degrade the quality of the CSI estimation. To deal with pilot contamination problem, a low-complexity subspace minimum mean square error (MMSE) estimation method is proposed in this paper. Specifically, our approach operates the MMSE estimation in a low-dimensional subspace to avoid large matrix manipulation. Meanwhile, subspace projection helps to discriminate the desired signal and interfering signals in the power domain. Interference analysis shows the MMSE estimation can achieve interference-free estimation even in a low-dimensional subspace with a large number of BS antennas, and non-overlapping angles of arrival (AoAs) between desired and interfering UEs. Furthermore, thanks to the low-rank property of the channel covariance matrix in massive MU-MIMO systems, a two-step covariance matrix subspace projection method is proposed for further computational complexity reduction. The complexity analysis and simulation results indicate that our proposed approach has better channel estimation accuracy with lower complexity than the conventional MMSE estimation when the number of BS antennas is large.

DOA estimation via ULA with mutual coupling in the presence of non-uniform noise

It is well known that classic direction-of-arrival (DOA) estimation methods yield poor accuracy and low resolution in the presence of mutual coupling and non-uniform noise. To tackle this problem, a DOA estimation method for jointly optimizing noise-free covariance matrix and mutual coupling coefficient is proposed in this paper. Based on the least squares (LS) criterion, the signal subspace and noise-free covariance matrix under mutual coupling can be obtained in an iterative manner. With the determined signal/noise subspace, the mutual coupling matrix can be reconstructed and the covariance one is estimated after mutual coupling compensation. Finally, the noise-free covariance matrix is preprocessed by spatial smoothing strategy and then used to determine the DOAs with the help of traditional DOA estimators. Compared to the conventional DOA estimation methods, simulation results demonstrate that, the proposed method can suppress the non-uniform noise significantly, alleviate the effect of mutual coupling on spatial smoothing technique obviously, as well as improve the DOA estimation performance in the case of coherent signals considerably.