Two-sided Orthogonal Decompositions Research Articles

Efficient Low-Rank Approximation of Matrices Based on Randomized Pivoted Decomposition

Given a matrix $\bf A$ with numerical rank $k$ , the two-sided orthogonal decomposition (TSOD) computes a factorization ${\bf A} = {\bf UDV}^T$ , where ${\bf U}$ and ${\bf V}$ are orthogonal, and ${\bf D}$ is (upper/lower) triangular. TSOD is rank-revealing as the middle factor ${\bf D}$ reveals the rank of $\bf A$ . The computation of TSOD, however, is demanding. In this paper, we present an algorithm called randomized pivoted TSOD (RP-TSOD), where the middle factor is lower triangular. Key in our work is the exploitation of randomization, and RP-TSOD is primarily devised to efficiently construct an approximation to a low-rank matrix. We provide three different types of bounds for RP-TSOD: (i) we furnish upper bounds on the error of the low-rank approximation, (ii) we bound the $k$ approximate principal singular values, and (iii) we derive bounds for the canonical angles between the approximate and the exact singular subspaces. Our bounds describe the characteristics and behavior of our proposed algorithm. Through numerical tests, we show the effectiveness of the devised bounds as well as our proposed algorithm.

An Alternative Algorithm for a Sliding Window ULV Decomposition

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The ULVD can be modified much faster than the SVD. When modifiying the ULVD, the accurate computation of the subspaces is required in certain time varying applications in signal processing. In this paper, we present an updating algorithm which is suitable for large scaled matrices of low rank and as effective as alternatives. The algorithm runs in O(n2) time.

Computing ULV Decomposition by Divide Conquer Method

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the Singular Value Decomposition (SVD). The ULVD can be modified much faster than the SVD. The accurate computation of the subspaces is required in applications in signal processing. In this study we introduce a divide conquer ULVD algorithm.

An Alternative Algorithm for the Refinement of ULV Decompositions

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). It is useful in applications of the SVD such as principal components where we are interested in approximating a matrix by one of lower rank. It can be updated and downdated much more quickly than an SVD. In many instances, the ULVD must be refined to improve the approximation it gives for the important right singular subspaces or to improve the matrix approximation. Present algorithms to perform this refinement require O(mn) operations if the rank of the matrix is k, where k is very close to 0 or n, but these algorithms require O(m n2) operations otherwise. Presented here is an alternative refinement algorithm that requires O(mn) operations no matter what the rank is. Our tests show that this new refinement algorithm produces similar improvement in matrix approximation and in the subspaces. We also propose slight improvements on the error bounds on subspaces and singular values computed by the ULVD.

A modified Gram–Schmidt-based downdating technique for ULV decompositions with applications to recursive TLS problems

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The ULVD can be updated and downdated much faster than the SVD, hence its utility in the solution of recursive total least squares (TLS) problems. However, the robust implementation of ULVD after the addition and deletion of rows (called updating and downdating, respectively) is not altogether straightforward. When updating or downdating the ULVD, the accurate computation of the subspaces necessary to solve the TLS problem is of great importance. In this paper, algorithms are given to compute simple parameters that can often show when good subspaces have been computed.

An efficient rank detection procedure for modifying the ULV decomposition

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The problem of adding and deleting rows from the ULVD (called updating and downdating, respectively) is considered. The ULVD can be updated and downdated much faster than the SVD, hence its utility.

Perturbation Analysis for Two-Sided (or Complete) Orthogonal Decompositions

Two-sided (or complete) orthogonal decompositions are good alternatives to the singular value decomposition (SVD) because they can yield good approximations to the fundamental subspaces associated with a numerically rank-deficient matrix. In this paper we derive perturbation bounds for the subspaces associated with a general two-sided orthogonal decomposition of a numerically rank-deficient matrix. The results imply the subspaces are only slightly more sensitive to perturbations than singular subspaces, provided the norm of the off-diagonal blocks of the middle matrices are sufficiently small with respect to the size of the perturbation. We consider regularizing the solution to the ill-conditioned least squares problem by truncating the decomposition and present perturbation theory for the minimum norm solution of the resulting least squares problem. The main results can be specialized to well known SVD-based perturbation bounds for singular subspaces as well as the truncated least squares solution.

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.

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.

An efficient Total Least Squares algorithm based on a rank-revealing two-sided orthogonal decomposition

Solving Total Least Squares (TLS) problemsAX≈B requires the computation of the noise subspace of the data matrix [A;B]. The widely used tool for doing this is the Singular Value Decomposition (SVD). However, the SVD has the drawback that it is computationally expensive. Therefore, we consider here a different so-called rank-revealing two-sided orthogonal decomposition which decomposes the matrix into a product of a unitary matrix, a triangular matrix and another unitary matrix in such a way that the effective rank of the matrix is obvious and at the same time the noise subspace is exhibited explicity. We show how this decompsition leads to an efficient and reliable TLS algorithm that can be parallelized in an efficient way.