URV Decomposition Research Articles

Randomized two-sided subspace iteration for low-rank matrix and tensor decomposition

The low-rank approximation of big data matrices and tensors plays a pivotal role in many modern applications. Although, a truncated version of the singular value decomposition (SVD) furnishes the best approximation, its computation is challenging on modern, multicore architectures. Recently, the randomized subspace iteration has shown to be a powerful tool in approximating large-scale matrices. In this paper we present a two-sided variant of the randomized subspace iteration. Novel in our work is the exploitation of the unpivoted QR factorization, rather than the SVD, for factorizing the compressed matrix. Hence our algorithm is a randomized rank-revealing URV decomposition. We prove the rank-revealingness of our algorithm by establishing bounds for the singular values as well as the other blocks of the compressed matrix. We further provide bounds on the error of the low-rank approximations of the proposed algorithm, in both 2- and Frobenius norm. In addition, we employ the proposed algorithm to efficiently compute low rank tensor decompositions: we present two randomized algorithms, one for the truncated higher-order SVD, and the other for the tensor SVD. We conduct numerical tests on (i) various classes of matrices, and (ii) synthetic tensors and real datasets to demonstrate the efficacy of the proposed algorithms.

A watermarking scheme for dual‐color images based on URV decomposition and image correction

In view of the fact that many watermarking schemes cannot well adapt to the occasion of image copyright protection with a large amount of information, and the copyright protection of color image is facing severe challenges, a dual-color image watermarking scheme based on URV decomposition (URVD) and image correction is designed in this paper. The two-dimensional Logistic-adjusted-Sine map is first used to generate two chaotic sequences for pixel-level scrambling and bit-level diffusion of the watermark image, and URVD is applied to watermarking scheme. After fully considering the relationship between image channels, the watermark is embedded into the maximum URVD coefficient by using quantization index modulation based on variable quantization steps. The greatest contribution of this paper is to propose a new image correction algorithm as a pretreatment of watermark extraction, which mainly includes fast detection of feature points, image correction, and sawtooth processing, and it solves some problems existing in image correction algorithms based on geometric characteristics. Experimental results show that the proposed scheme has good invisibility, strong robustness, and high security. Compared with existing schemes, the proposed scheme has better performance under the same actual embedding rate, especially stronger and more effective robustness against some common geometric attacks.

A Fast and Efficient Sliding Window based URV Decomposition Algorithm for Template Tracking

A Fast and Efficient Sliding Window based URV Decomposition Algorithm for Template Tracking

Robust Update Algorithms for Zero-Forcing Detection in Uplink Large-Scale MIMO Systems

A zero-forcing (ZF) detector is proposed for the uplink channel of large-scale multiple-input-multiple-output systems based on the URV decomposition of the channel matrix. Two efficient algorithms are also proposed for updating and downdating the channel decomposition matrices upon a user’s arrival or exit of the cell, respectively, which avoids re-calculating the URV decomposition. Specifically, the decomposition is updated through the Gram–Schmidt procedure when a user equipment (UE) joins the cell, and is downdated through a set of Givens rotations when a UE leaves the cell. The computational complexity of the proposed detector (referred to as the URV-ZF detector) is evaluated and compared with that of existing methods. It is shown through numerical studies that the URV-ZF detector achieves a lower bit error rate than existing schemes with a slight increase in the computational complexity.

Updating approximate principal components with applications to template tracking

SummaryAdaptive principal component analysis is prohibitively expensive when a large‐scale data matrix must be updated frequently. Therefore, we consider the truncated URV decomposition that allows faster updates to its approximation to the singular value decomposition while still producing a good enough approximation to recover principal components. Specifically, we suggest an efficient algorithm for the truncated URV decomposition when a rank 1 matrix updates the data matrix. After the algorithm development, the truncated URV decomposition is successfully applied to the template tracking problem in a video sequence proposed by Matthews et al. [IEEE Trans. Pattern Anal. Mach. Intell., 26:810‐815 2004], which requires computation of the principal components of the augmented image matrix at every iteration. From the template tracking experiments, we show that, in adaptive applications, the truncated URV decomposition maintains a good approximation to the principal component subspace more efficiently than other procedures.

Structure-preserving Computation of Stable Deflating Subspaces

Structure-preserving numerical techniques for computation of stable deflating subspaces, with applications in control systems design, are presented. The techniques use extended skew-Hamiltonian/Hamiltonian matrix pencils, and specialized algorithms to exploit their structure: the symplectic URV decomposition, periodic QZ algorithm, solution of periodic Sylvester-like equations, etc. The structure-preserving approach has the potential to avoid the numerical difficulties which are encountered for a traditional, non-structured solution, returned by the currently available software tools.

An algorithm for rank estimation and subspace tracking

This article presents an URV-based matrix decomposition, the truncated URV decomposition, and an updating algorithm for it. The complexity of the updating is for an m-by- n matrix of rank r. The theoretical and numerical results presented shows that the decomposition can be a good alternative to the singular value decomposition.

Blind Channel Estimation Based on URV Decomposition Technique for Uplink of MC-CDMA

In this paper, we investigate a blind channel estimation method for Multi-carrier CDMA systems that use a subspace decomposition technique. This technique exploits the orthogonality property between the noise subspace and the received user codes to obtain channel of each user. In the past we used Singular Value Decomposition (SVD) technique but SVD have most computational complexity so in this paper use a new algorithm called URV Decomposition, which serve as an intermediary between the QR decomposition and SVD, replaced in SVD technique to track the noise space of the received data. Because of the URV decomposition has almost the same estimation performance as the SVD, but has less computational complexity. Keywords—Channel estimation, MC-CDMA, SVD, URV

On the Computation of the Restricted Singular Value Decomposition via the Cosine-Sine Decomposition

In this paper, we show that the restricted singular value decomposition of a matrix triplet $A\in \R^{n \times m}, B\in \R^{n \times l}, C\in \R^{p \times m}$ can be computed by means of the cosine-sine decomposition. In the first step, the matrices A, B, C are reduced to a lower-dimensional matrix triplet ${\cal A}, {\cal B}, {\cal C}$, in which ${\cal B}$ and ${\cal C}$ are nonsingular, using orthogonal transformations such as the QR-factorization with column pivoting and the URV decomposition. In the second step, the components of the restricted singular value decomposition of A, B, C are derived from the singular value decomposition of ${\cal B}^{-1}{\cal A}{\cal C}^{-1}$. Instead of explicitly forming the latter product, a link with the cosine-sine decomposition, which can be computed by Van Loan's method, is exploited. Some numerical examples are given to show the performance of the presented method.

Updating a Generalized URV Decomposition

An updating scheme for a quotient type generalization of a URV decomposition of two matrices is introduced. This decomposition allows low complexity updating as rows are added to two rectangular matrices, determining the dimension of three distinct subspaces. One of these subspaces is the intersection of the range space of the two matrices---information which leads to a potential application in subspace algorithms for system identification.

UTV Tools: Matlab Templates for Rank-Revealing UTV Decompositions

We describe a Matlab 5.2 package for computing and modifying certain rank-revealing decompositions that have found widespread use in signal processing and other applications. The package focuses on algorithms for URV and ULV decompositions, collectively known as UTV decompositions. We include algorithms for the ULLV decomposition, which generalizes the ULV decomposition to a pair of matrices. For completeness a few algorithms for computation of the RRQR decomposition are also included. The software in this package can be used as is, or can be considered as templates for specialized implementations on signal processors and similar dedicated hardware platforms.

Parallel codes for computing the numerical rank

In this paper we present an experimental comparison of several numerical tools for computing the numerical rank of dense matrices. The study includes the well-known SVD, the URV decomposition, and several rank-revealing QR factorizations: the QR factorization with column pivoting, and two QR factorizations with restricted column pivoting. Two different parallel programming methodologies are analyzed in our paper. First, we present block-partitioned algorithms for the URV decomposition and rank-revealing QR factorizations which provide efficient implementations on shared memory environments. Furthermore, we also present parallel distributed algorithms, based on the message-passing paradigm, for computing rank-revealing QR factorizations on multicomputers.

Bounding the Subspaces from Rank Revealing Two-Sided Orthogonal Decompositions

The singular value decomposition (SVD) is a widely used computational tool in various applications. However, in some applications the SVD is viewed as computationally demanding or difficult to update. The rank revealing QR (RRQR) decomposition and the recently proposed URV and ULV decompositions are promising alternatives for determining the numerical rank k of an $m \times n$ matrix and approximating its fundamental numerical subspaces whenever $k \approx \min (m,n)$. In this paper we prove a posteriori bounds for assessing the quality of the subspaces obtained by two-sided orthogonal decompositions. In particular, we show that the quality of the subspaces obtained by the URV or ULV algorithm depends on the quality of the condition estimator and not on a gap condition. From our analysis we conclude that these decompositions may be more accurate alternatives to the SVD than the RRQR decomposition. Finally, we implement the algorithms in an adaptive manner, which is particularly useful for applications where the “noise” subspace must be computed, such as in signal processing or total least squares.

Analysis of a QR Algorithm for Computing Singular Values

We extend the Golub--Kahan algorithm for computing the singular value decomposition of bidiagonal matrices to triangular matrices $R$. Our algorithm avoids the explicit formation of $R^TR$ or $RR^T$. We derive a relation between left and right singular vectors of triangular matrices and use it to prove monotonic convergence of singular values and singular vectors. The convergence rate for singular values equals the square of the convergence rate for singular vectors. The convergence behaviour explains the occurrence of deflation in the interior of the matrix. We analyse the relationship between our algorithm and rank-revealing QR and URV decompositions. As a consequence, we obtain an algorithm for computing the URV decomposition, as well as a divide-and-conquer algorithm that computes singular values of dense matrices and may be beneficial on a parallel architecture. Our perturbation result for the smallest singular values of a triangular matrix is stronger than the traditional results because it guarantees high relative accuracy in the smallest singular values after an off-diagonal block of the matrix has been set to zero.

On the stability of sequential updates and downdates

The updating and downdating of Cholesky decompositions has important applications in a number of areas. There is essentially one standard updating algorithm, based on plane rotations, which is backward stable. Three downdating algorithms have been treated in the literature: the LINPACK algorithm, the method of hyperbolic transformations, and Chambers' (1971) algorithm. Although none of these algorithms is backward stable, the first and third satisfy a relational stability condition. It is shown that relational stability extends to a sequence of updates and downdates. In consequence, other things being equal, if the final decomposition in the sequence is well conditioned, it will be accurately computed, even though intermediate decompositions may be almost completely inaccurate. These results are also applied to the two-sided orthogonal decompositions, such as the URV decomposition.

Downdating the Rank-Revealing URV Decomposition

An accurate algorithm is presented for downdating a row in the rank-revealing URV decomposition that was recently introduced by Stewart. By downdating the full rank part and the noise part in two separate steps, the new algorithm can produce accurate results even for ill-conditioned problems. Such problems occur, for example, when the rank of the matrix is decreased as a consequence of the downdate. Other possible generalizations of existing QR decomposition downdating algorithms for the rank-revealing URV downdating are discussed. Numerical test results are presented that compare the performance of these new URV decomposition downdating algorithms in the sliding window method.

URV ESPRIT for tracking time-varying signals

ESPRIT is an algorithm for determining the fixed directions of arrival of a set of narrowband signals at an array of sensors. Unfortunately, its computational burden makes it unsuitable for real time processing of signals with time-varying directions of arrival. The authors develop a new implementation of ESPRIT that has potential for real time processing. It is based on a rank-revealing URV decomposition, rather than the eigendecomposition or singular value decomposition used in previous ESPRIT algorithms. The authors demonstrate its performance on simulated data representing both constant and time-varying signals. They find that the URV-based ESPRIT algorithm is effective for estimating time-varying directions-of-arrival at considerable computational savings over the SVD-based algorithm. >

Updating URV decompositions in parallel

A URV decomposition of a matrix is a factorization of the matrix into the product of a unitary matrix (U), an upper triangular matrix (R), and another unitary matrix (V). In [8] it was shown how to update a URV decomposition in such a way that it reveals the effective rank of the matrix. It was also argued that the updating procedure could be implemented in parallel on a linear array of processors; however, no specific algorithms were given. This paper gives a detailed implementation of the updating procedure.

On the QR algorithm and updating the SVD and the URV decomposition in parallel

A Jacobi-type updating algorithm for the SVD or the URV decomposition is developed, which is related to the QR algorithm for the symmetric eigenvalue problem. The algorithm employs one-sided transformations, and therefore provides a cheap alternative to earlier-developed updating algorithms based on two-sided transformations. The present algorithm as well as the corresponding systolic implementation is therefore rpughly twice as fast as the former method, while the tracking properties are preserved. The algorithm is also extended to the two-matrix QSVD or QURV case. Finally, the differences with a number of closely related algorithms that have recently been proposed are discussed.

An updating algorithm for subspace tracking

In certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components. The usual tool for doing this is the singular value decomposition. However, the singular value decomposition has the drawback that it requires O(p/sup 3/) operations to recompute when a new sample arrives. It is shown that a different decomposition, called the URV decomposition, is equally effective in exhibiting the null space and can be updated in O(p/sup 2/) time. The updating technique can be run on a linear array of p processors in O(p) time.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>