QR decomposition

About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.

Downdating the Rank-Revealing URV Decomposition

Abstract: 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.

LWT- QR decomposition based robust and efficient image watermarking scheme using Lagrangian SVR

Abstract: In this paper, an efficient and robust image watermarking scheme based on lifting wavelet transform (LWT) and QR decomposition using Lagrangian support vector regression (LSVR) is presented. After performing one level decomposition of host image using LWT, the low frequency subband is divided into 4?×?4 non-overlapping blocks. Based on the correlation property of lifting wavelet coefficients, each selected block is followed by QR decomposition. The significant element of first row of R matrix of each block is set as target to LSVR for embedding the watermark. The remaining elements (called feature vector) of upper triangular matrix R act as input to LSVR. The security of the watermark is achieved by applying Arnold transformation to original watermark to get its scrambled image. This scrambled image is embedded into the output (predicted value) of LSVR compared with the target value using optimal scaling factor to reduce the tradeoff between imperceptibility and robustness. Experimental results show that proposed scheme not only efficient in terms of computational cost and memory requirement but also achieve good imperceptibility and robustness against image processing operations compared to the state-of-art techniques.

Matrix-Group Algorithm via Improved Whitening Process for Extracting Statistically Independent Sources From Array Signals

Abstract: This paper addresses the problem of blind separation of multiple independent sources from observed array output signals. The main contributions in this paper include an improved whitening scheme for estimation of signal subspace, a novel biquadratic contrast function for extraction of independent sources, and an efficient alterative method for joint implementation of a set of approximate diagonalization-structural matrices. Specifically, an improved whitening scheme is first developed by estimating the signal subspace jointly from a set of diagonalization-structural matrices based on the proposed cyclic maximizer of an interesting cost function. Moreover, the globally asymptotical convergence of the proposed cyclic maximizer is analyzed and proved. Next, a novel biquadratic contrast function is proposed for extracting one single independent component from a slice matrix group of any order cumulant of the array signals in the presence of temporally white noise. A fast fixed-point algorithm that is a cyclic minimizer is constructed for searching a minimum point of the proposed contrast function. The globally asymptotical convergence of the proposed fixed-point algorithm is analyzed. Then, multiple independent components are obtained by using repeatedly the proposed fixed-point algorithm for extracting one single independent component, and the orthogonality among them is achieved by the well-known QR factorization. The performance of the proposed algorithms is illustrated by simulation results and is compared with three related blind source separation algorithms

Dynamical system analyzer

Abstract: A dynamical system analyser (10) incorporates a computer (22) to perform a singular value decomposition of a time series of signals from a nonlinear (possibly chaotic) dynamical system (14). Relatively low-noise singular vectors from the decomposition are loaded into a finite impulse response filter (34). The time series is formed into Takens' vectors each of which is projected onto each of the singular vectors by the filter (34). Each Takens' vector thereby provides the co-ordinates of a respective point on a trajectory of the system (14) in a phase space. A heuristic processor (44) is used to transform delayed co-ordinates by QR decomposition and least squares fitting so that they are fitted to non-delayed co-ordinates. The heuristic processor (44) generates a mathematical model to implement this transformation, which predicts future system states on the basis of respective current states. A trial system is employed to generate like co-ordinates for transformation in the heuristic processor (44). This produces estimates of the trial system's future states predicted from the comparison system's model. Alternatively, divergences between such estimates and actual behavior may be obtained. As a further alternative, mathematical models derived by the analyser (10) from different dynamical systems may be compared.

Local Convergence Analysis of FastICA and Related Algorithms

Abstract: The FastICA algorithm is one of the most prominent methods to solve the problem of linear independent component analysis (ICA). Although there have been several attempts to prove local convergence properties of FastICA, rigorous analysis is still missing in the community. The major difficulty of analysis is because of the well-known sign-flipping phenomenon of FastICA, which causes the discontinuity of the corresponding FastICA map on the unit sphere. In this paper, by using the concept of principal fiber bundles, FastICA is proven to be locally quadratically convergent to a correct separation. Higher order local convergence properties of FastICA are also investigated in the framework of a scalar shift strategy. Moreover, as a parallelized version of FastICA, the so-called QR FastICA algorithm, which employs the QR decomposition (Gram-Schmidt orthonormalization process) instead of the polar decomposition, is shown to share similar local convergence properties with the original FastICA.