In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

Anomalies are unusual and significant changes in a network's traffic levels, which can often span multiple links. Diagnosing anomalies is critical for both network operators and end users. It is a difficult problem because one must extract and interpret anomalous patterns from large amounts of high-dimensional, noisy data.

. Let A 2 R m\Thetan be a matrix with known singular values and singular vectors, and let A 0 be the matrix obtained by appending a row to A. We present stable and fast algorithms for computing the singular values and the singular vectors of A 0 in O \Gamma (m + n) min(m;n) log 2 2 ffl \Delta floating point operations, where ffl is the machine precision. Previous algorithms can be unstable and compute the singular values and the singular vectors of A 0 in O \Gamma (m + n) min 2 (m;n) \Delta floating point operations. 1. Introduction. The singular value decomposition (SVD) of a matrix A 2 R m\Thetan is A = U\Omega V T ; (1.1) where U 2 R m\Thetam and V 2 R n\Thetan are orthonormal; and\Omega 2 R m\Thetan is zero except on the main diagonal, which has non-negative entries in decreasing order. The columns of U and V are the left singular vectors and the right singular vectors of A, respectively; the diagonal entries of\Omega are the singular values of A....

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Modified Jacobi and Kogbetliantz algorithms are derived by combining methods for modifying the orthogonal rotations. These methods are characterized by the use of approximate orthogonal rotations and the factorization of these rotations. The presented new approximations exhibit better properties and require less computational cost than known approximations. Suitable approximations are applied together with factorized rotation schemes in order to gain square root free or square root and division free algorithms. The resulting approximate and factorized rotation schemes are highly suited for parallel implementations. The convergence of the algorithms is analyzed and an application in signal processing is discussed.

In this paper a Generalised Singular Value Decomposition (GSVD) based algorithm is proposed for enhancing multi-microphone speech signals degraded by additive coloured noise. This GSVD-based multi-microphone speech signal...

In this paper, two adaptive algorithms are presented for robust time delay estimation (TDE) in acoustic environments where a large amount of background noise and reverberation is present. Recently, an adaptive eigenvalue decomposition (EVD) algorithm has been developed for TDE between two microphones in highly reverberant acoustic environments. In this paper, we extend the adaptive EVD algorithm to noisy and reverberant acoustic environments, by deriving an adaptive stochastic gradient algorithm for the generalised eigenvalue decomposition (GEVD) or by prewhitening the noisy microphone signals. In addition, we extend all considered TDE algorithms to the case of more than two microphones. We have performed

In this paper a parallel implementation of the SVD--updating algorithm using approximate rotations is presented. In its original form the SVD--updating algorithm had numerical problems if no reorthogonalization steps were applied. Representing the orthogonal matrix V (right singular vectors) using its parameterization in terms of the rotation angles of n(n \Gamma 1)=2 plane rotations these reorthogonalization steps can be avoided during the SVD-updating algorithm [18]. This results in a SVD--updating algorithm where all computations (matrix vector multiplication, QRD--updating, Kogbetliantz's algorithm) are entirely based on the evaluation and application of orthogonal plane rotations. Therefore, in this form the SVD--updating algorithm is amenable to an implementation using CORDIC--based approximate rotations. Using CORDIC--based approximate rotations the n(n\Gamma1)=2 rotations representing V (as well as all other rotations) are only computed to a certain approximation accuracy (in the basis arctan 2 i). All necessary computations required during the SVD--updating algorithm (exclusively rotations) are executed with the same accuracy, i.e., only r � � w (w: wordlength) elementary orthonormal ��--rotations are used per plane rotation. Simulations show the efficiency of the implementation using CORDIC--based approximate rotations.

In this paper, we propose new algorithms for approximate updating of the singular value decomposition (SVD) of an exponentially weighted data matrix after appending a new row. The algorithms are obtained in two steps: noise subspace sphericalization is first used to deflate the problem; the right singular vectors and the singular values are then efficiently updated by means of a recently proposed constrained perturbation approach. The latter is based on Givens rotations and thus preserves the orthonormality of the updated singular vectors. The new algorithms have complexity ranging from O(Nr) to O(Nr 2 ), where N and r respectively denote the data vector and signal-subspace dimensions. Their convergence behavior in subspace tracking applications is investigated by means of the ODE method and the results are supported by computer experiments. 1. Introduction Subspace-based signal analysis methods have proven to be of great utility in a wide variety of detection and parameter estimat...

This paper presents efficient Schur-type algorithms for estimating the column space (signal subspace) of a low rank data matrix corrupted by additive noise. Its computational structure and complexity are similar to that of an LQ-decomposition, except for the fact that plane and hyperbolic rotations are used. Therefore, they are well suited for a parallel (systolic) implementation. The required rank decision, i.e., an estimate of the number of signals, is automatic, and updating as well as downdating are straightforward. The new scheme computes a matrix of minimal rank which is γ-close to the data matrix in the matrix 2-norm, where γ is a threshold that can be determined from the noise level. Since the resulting approximation error is not minimized, critical scenarios lead to a certain loss of accuracy compared to SVD-based methods. This loss of accuracy is compensated by using Unitary ESPRIT in conjunction with the Schur-type subspace estimation scheme. Unitary ESPRIT represents a simple way to constrain the estimated