Column Subspace Research Articles

Channel Estimation for Reconfigurable Intelligent Surface Aided Multi-User mmWave MIMO Systems

Channel acquisition is one of the main challenges for the deployment of reconfigurable intelligent surface (RIS) aided communication systems. This is because an RIS has a large number of reflective elements, which are passive devices with no active transmitting/receiving abilities. In this paper, we study the channel estimation problem for the RIS aided multi-user millimeter-wave (mmWave) multi-input multi-output (MIMO) system. Specifically, we propose a novel channel estimation protocol for the above system to estimate the cascaded channels, which are the products of the channels from the base station (BS) to the RIS and from the RIS to the users. Further, since the cascaded channels are typically sparse, this allows us to formulate the channel estimation problem as a sparse recovery problem using compressive sensing (CS) techniques, thereby allowing the channels to be estimated with less training overhead. Moreover, the sparse channel matrices of the cascaded channels of all users have a common block sparsity structure due to the common channel between the BS and the RIS. To take advantage of the common sparsity pattern, we propose a two-step multi-user joint channel estimation procedure. In the first step, we make use of the common column-block sparsity and project the received signals onto the common column subspace. In the second step, we make use of the row-block sparsity of the projected signals and propose a multi-user joint sparse matrix recovery algorithm that takes into account the common channel between the BS and the RIS.

Efficiency of subspace-based estimators for elliptical symmetric distributions

Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameter-dependent subspace have proved very useful in many applications in signal processing. The purpose of this paper is to complement theoretical results already available on the asymptotic (in the number of measurements) performance of subspace-based estimators derived in the Gaussian context to real elliptical symmetric (RES), circular complex elliptical symmetric (C-CES) and non-circular CES (NC-CES) distributed observations in the same framework. First, the asymptotic distribution of M-estimates of the orthogonal projection matrix is derived from those of the M-estimates of the covariance matrix. This allows us to characterize the asymptotically minimum variance (AMV) estimator based on estimates of orthogonal projectors associated with different M-estimates of the covariance matrix. A closed-form expression is then given for the AMV bound on the parameter of interest characterized by the column subspace of the mixing matrix of general linear mixture models. We also specify the conditions under which the AMV bound based on Tyler’s M-estimate attains the stochastic Cramér-Rao bound (CRB) for the complex Student t and complex generalized Gaussian distributions. Finally, we prove that the AMV bound attains the stochastic CRB in the case of maximum likelihood (ML) M-estimate of the covariance matrix for RES, C-CES and NC-CES distributed observations, which is equal to the semiparametric CRB (SCRB) recently introduced.

A Sequential Subspace Method for Millimeter Wave MIMO Channel Estimation

Data transmission over the mmWave in fifth-generation wireless networks aims to support very high speed wireless communications. A substantial increase in spectrum efficiency for mmWave transmission can be achieved by using advanced hybrid precoding, for which accurate channel state information is the key. Rather than estimating the entire channel matrix, directly estimating subspace information, which contains fewer parameters, does have enough information to design transceivers. However, the large channel use overhead and associated computational complexity in the existing channel subspace estimation techniques are major obstacles to deploy the subspace approach for channel estimation. In this paper, we propose a sequential two-stage subspace estimation method that can resolve the overhead issues and provide accurate subspace information. Utilizing a sequential method enables us to avoid manipulating the entire high-dimensional training signal, which greatly reduces the complexity. Specifically, in the first stage, the proposed method samples the columns of channel matrix to estimate its column subspace. Then, based on the obtained column subspace, it optimizes the training signals to estimate the row subspace. For a channel with $N_r$ receive antennas and $N_t$ transmit antennas, our analysis shows that the proposed technique only requires $O(N_t)$ channel uses, while providing a guarantee of subspace estimation accuracy. By theoretical analysis, it is shown that the similarity between the estimated subspace and the true subspace is linearly related to the signal-to-noise ratio (SNR), i.e., $O(\text{SNR})$, at high SNR, while quadratically related to the SNR, i.e., $O(\text{SNR}^2)$, at low SNR. Simulation results show that the proposed sequential subspace method can provide improved subspace accuracy, normalized mean squared error, and spectrum efficiency over existing methods.

Mixed-Field Source Localization Based on the Non-Hermitian Matrix

In this paper, an efficient high-order multiple signal classification (MUSIC)-like method is proposed for mixed-field source localization. Firstly, a non-Hermitian matrix is designed based on a high-order cumulant. One of the steering matrices, that is related only with the directions of arrival (DOA), is proved to be orthogonal with the eigenvectors corresponding to the zero eigenvalues. The other steering matrix that contains the information of both the DOA and range is proved to span the same column subspace with the eigenvectors corresponding to the non-zero eigenvalues. By applying the Gram–Schmidt orthogonalization, the range estimation can be achieved one by one after substituting each estimated DOA. The analysis shows that the computational complexity of the proposed method is lower than other methods, and the effectiveness of the proposed method is shown with some simulation results.

Leveraging subspace information for low-rank matrix reconstruction

The problem of low-rank matrix reconstruction arises in various applications in communications and signal processing. The state of the art research largely focuses on the recovery techniques that utilize affine maps satisfying the restricted isometry property (RIP). However, the affine map design and reconstruction under a priori information, i.e., column or row subspace information, has not been thoroughly investigated. To this end, we present designs of affine maps and reconstruction algorithms that fully exploit the low-rank matrix subspace information. Compared to the randomly generated affine map, the proposed affine map design permits an enhanced reconstruction. In addition, we derive an optimal representation of low-rank matrices, which is exploited to optimize the rank and subspace of the estimate by adapting them to the noise level in order to achieve the minimum mean square error (MSE). Moreover, in the case when the subspace information is not a priori available, we propose a two-step algorithm, where, in the first step, it estimates the column subspace of a low-rank matrix, and in the second step, it exploits the estimated information to complete the reconstruction. The simulation results show that the proposed algorithm achieves robust performance with much lower complexity than existing reconstruction algorithms.

COSCEB: Comprehensive search for column-coherent evolution biclusters and its application to hub gene identification

Biclustering is an increasingly used data mining technique for searching groups of co-expressed genes across the subset of experimental conditions from the gene-expression data. The group of co-expressed genes is present in the form of various patterns called a bicluster. A bicluster provides significant insights related to the functionality of genes and plays an important role in various clinical applications such as drug discovery, biomarker discovery, gene network analysis, gene identification, disease diagnosis, pathway analysis etc. This paper presents a novel unsupervised approach 'COmprehensive Search for Column-Coherent Evolution Biclusters (COSCEB)' for a comprehensive search of biologically significant column-coherent evolution biclusters. The concept of column subspace extraction from each gene pair and Longest Common Contiguous Subsequence (LCCS) is employed to identify significant biclusters. The experiments have been performed on both synthetic as well as real datasets. The performance of COSCEB is evaluated with the help of key issues. The issues are comprehensive search, Deep OPSM bicluster, bicluster types, bicluster accuracy, bicluster size, noise, overlapping, output nature, computational complexity and biologically significant biclusters. The performance of COSCEB is compared with six all-time famous biclustering algorithms SAMBA, OPSM, xMotif, Bimax, Deep OPSM- and UniBic. The result shows that the proposed approach performs effectively on most of the issues and extracts all possible biologically significant column-coherent evolution biclusters which are far more than other biclustering algorithms. Along with the proposed approach, we have also presented the case study which shows the application of significant biclusters for hub gene identification.

A Randomized Subspace Learning Based Anomaly Detector for Hyperspectral Imagery

This paper proposes a randomized subspace learning based anomaly detector (RSLAD) for hyperspectral imagery (HSI). Improved from robust principal component analysis, the RSLAD assumes that the background matrix is low-rank, and the anomaly matrix is sparse with a small portion of nonzero columns (i.e., column-wise). It also assumes the anomalies do not lie in the column subspace of the background and aims to find a randomized subspace of the background to detect the anomalies. First, random techniques including random sampling and random Hadamard projections are implemented to construct a coarse randomized columns subspace of the background with reduced computational cost. Second, anomaly columns are searched and removed from the coarse randomized column subspace by solving a series of least squares problems, resulting in a purified randomized column subspace. Third, the nonzero columns in the anomaly matrix are located by projecting all the pixels on the orthogonal subspace of the purified subspace, and the anomalies are finally detected based on the L2 norm of the columns in the anomaly matrix. The detection performance of RSLAD is compared with four state-of-the-art methods, including global Reed-Xiaoli (GRX), local RX (LRX), collaborative-representation based detector (CRD), and low-rank and sparse matrix decomposition base anomaly detector (LRaSMD). Experimental results show good detection performance of RSLAD with lower computational cost. Therefore, the proposed RSLAD offers an alternative option for hyperspectral anomaly detection.

Randomized subspace-based robust principal component analysis for hyperspectral anomaly detection

A randomized subspace-based robust principal component analysis (RSRPCA) method for anomaly detection in hyperspectral imagery (HSI) is proposed. The RSRPCA combines advantages of randomized column subspace and robust principal component analysis (RPCA). It assumes that the background has low-rank properties, and the anomalies are sparse and do not lie in the column subspace of the background. First, RSRPCA implements random sampling to sketch the original HSI dataset from columns and to construct a randomized column subspace of the background. Structured random projections are also adopted to sketch the HSI dataset from rows. Sketching from columns and rows could greatly reduce the computational requirements of RSRPCA. Second, the RSRPCA adopts the columnwise RPCA (CWRPCA) to eliminate negative effects of sampled anomaly pixels and that purifies the previous randomized column subspace by removing sampled anomaly columns. The CWRPCA decomposes the submatrix of the HSI data into a low-rank matrix (i.e., background component), a noisy matrix (i.e., noise component), and a sparse anomaly matrix (i.e., anomaly component) with only a small proportion of nonzero columns. The algorithm of inexact augmented Lagrange multiplier is utilized to optimize the CWRPCA problem and estimate the sparse matrix. Nonzero columns of the sparse anomaly matrix point to sampled anomaly columns in the submatrix. Third, all the pixels are projected onto the complemental subspace of the purified randomized column subspace of the background and the anomaly pixels in the original HSI data are finally exactly located. Several experiments on three real hyperspectral images are carefully designed to investigate the detection performance of RSRPCA, and the results are compared with four state-of-the-art methods. Experimental results show that the proposed RSRPCA outperforms four comparison methods both in detection performance and in computational time.

A maximal projection solution of ill-posed linear system in a column subspace, better than the least squares solution

A better method than the least squares solution is proposed in this paper to solve ann-dimensional ill-posed linear equations system Ax=b in an m-dimensional column subspace Cm, which is selected in such a way that each column in Cm is in a closer proximity to b. We maximize the orthogonal projection of b onto y≔Ax to find an approximate solution x∈span{a,Cm}, where a is a nonzero free vector. Then, we can prove that the maximal projection solution (MP) is better than the least squares solution (LS) with ‖b−AxMP‖<‖b−AxLS‖. Numerical examples of inverse problems under a large noise maybe up to 30% are discussed which confirm the efficiency of presently developed MP algorithms: MPA and MPA(m).

The importance of the sequential synthesis methodology in the optimal distillation sequences design

The sequential design method is presented as a complementary tool of the systematical synthesis procedure that allows to define a clear connection among the different types of distillation column sequences. In particular, the connection with the simple column subspace is considered, since this subspace represents the comparison reference for all the alternatives considered. The sequential design procedure, based on the correspondence between the functionality of the column's section among the simple columns and the derived sequences, is compared with a mathematical based optimization algorithm. The separations of a four-component near ideal mixture and the azeotropic ethanol–water mixture are considered as case studies and the designs obtained applying both methods have been compared. The results confirmed that the sequential design method is a fast and reliable tool in the optimal design of the column sequence.

CGLS-GCV: a hybrid algorithm for low-rank-deficient problems

Given A=A+E∈ C m×n , where rank( A)⪡min( m, n), and b ̃ =b+ε , we investigate the following problems: (a) the construction of approximate minimum norm solutions of the least squares problem min‖ Ax− b‖, and (b) the computation of approximations of the column (row) subspace of A. We propose an algorithm for solving these problems based on conjugate gradient iterations followed by regularization in the generated Krylov subspace. Regularization is introduced for estimating rank( A) and implemented using the generalized cross-validation technique. We report the outcome of numerical experiments, showing that the new algorithm yields results with accuracy comparable to that of the SVD, but at a lower computational cost.

On the Successive Projections Approach to Least-Squares Problems

In this paper, we suggest a generalized Gauss–Seidel approach to sparse linear and nonlinear least-squares problems. The algorithm, closely related to one given by Elfving (1980), uses the work of Curtis, Powell, and Reid (1974) as extended by Coleman and Moré (1983) to divide the variables into nondisjoint groups of structurally orthogonal columns and then projects the updated residual into each column subspace of the Jacobian in turn. In the linear case, this procedure can be viewed as an alternate ordering of the variables in the Gauss–Seidel method. Preliminary tests indicate that this leads quickly to cheap solutions of limited accuracy for linear problems, and that this approach is promising for an inexact Gauss–Newton analog of the inexact Newton approach of Dembo, Eisenstat, and Steihaug (1982).