A Rank-Revealing Method with Updating, Downdating, and Applications. Part II

Lee, Tsung-Lin; Li, Tien-Yien; Zeng, Zhonggang

doi:10.1137/07068179x

Cited by 16 publications

(12 citation statements)

References 19 publications

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“…In this paper, we mainly employ RRQR, although we use τ -accurate SVD in presenting the basic idea. Many RRQR algorithms have been developed [17,18,19,31,38]. As an example, the QR with column pivoting method [17,28] computes the following factorization after k steps:…”

Section: Block Compressionmentioning

confidence: 99%

Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices

Xia¹,

Gu²

2010

SIAM J. Matrix Anal. & Appl.

125

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Given a symmetric positive definite matrix A, we compute a structured approximate Cholesky factorization A ≈ R T R up to any desired accuracy, where R is an upper triangular hierarchically semiseparable (HSS) matrix. The factorization is stable, robust, and efficient. The method compresses off-diagonal blocks with rank-revealing orthogonal decompositions. In the meantime, positive semidefinite terms are automatically and implicitly added to Schur complements in the factorization so that the approximation R T R is guaranteed to exist and be positive definite. The approximate factorization can be used as a structured preconditioner which does not break down. No extra stabilization step is needed. When A has an off-diagonal low-rank property, or when the off-diagonal blocks of A have small numerical ranks, the preconditioner is data sparse and is especially efficient. Furthermore, the method has a good potential to give satisfactory preconditioning bounds even if this low-rank property is not obvious. Numerical experiments are used to demonstrate the performance of the method. The method can be used to provide effective structured preconditioners for large sparse problems when combined with some sparse matrix techniques. The hierarchical compression scheme in this work is also useful in the development of more HSS algorithms.

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Section: Block Compressionmentioning

confidence: 99%

Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices

Xia¹,

Gu²

2010

SIAM J. Matrix Anal. & Appl.

125

View full text Add to dashboard Cite

show abstract

“…Indeed, numerical experiments indicate that the analogy between singular values of F and the absolute values of diagonal entries of D remains even for low rank d, strongly suggesting that the LDL * handles the nearly singular matrix F almost as well as the singular value decomposition does. These observations are not totally surprising because it is known that the LU algorithms, of which the LDL * is a special case, with suitable pivoting strategies, can reveal ranks properly (Hansen & Yalamov, 2001;Miranian & Gu, 2003;Kawabata et al, 2004;Lee et al, 2009;Foster & Liu, 2012).…”

Section: Numerical Experimentsmentioning

confidence: 94%

“…Even without the presence of noise, a nearly singular matrix may have to be treated as singular. Rank revealing algorithms developed thus far in the literature cannot resolve this problem perfectly (Lee et al, 2009;Foster & Liu, 2012).…”

Section: Recursive Formulamentioning

confidence: 99%

On the finite rank and finite-dimensional representation of bounded semi-infinite Hankel operators

Chu¹,

Lin

2014

IMA J Numer Anal

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Bounded, semi-infinite Hankel matrices of finite rank over the space 2 of square-summable sequences occur frequently in classical analysis and engineering applications. The notion of finite rank often appears under different contexts and the literature is diverse. The first part of this paper reviews some elegant, classical criteria and establishes connections among the various characterizations of finite rank in terms of rational functions, recursion, matrix factorizations and sinusoidal signals. All criteria require 2d parameters, though with different meanings, for a matrix of rank d. The Vandermonde factorization, in particular, permits immediately a singular-value preserving, finite-dimensional representation of the original semi-infinite Hankel matrix and, hence, makes it possible to retrieve the nonzero singular values of the semi-infinite Hankel matrix. The second part of this paper proposes using the LDL * decomposition of a specially constructed sample matrix to find the unitarily equivalent finite-dimensional representation. This approach enjoys several advantages, including the ease of computation by avoiding infinitedimensional vectors, the ability to reveal rank deficiency and the established pivoting strategy for stability. No error analysis is given, but several computational issues are discussed.

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“…Fortunately, only K left singular vectors are required in our proposed algorithm. The power method [19] is used to replace the complete SVD procedure to control the total computational complexity of the proposed algorithm.…”

Section: ) Single-cell Single-user Scenariomentioning

confidence: 99%

“…According to [19], as the recursive time i increases to infinity, p i converges to the right dominant singular vector v corresponding to the largest singular value.…”

Section: ) Single-cell Single-user Scenariomentioning

confidence: 99%

Low-Complexity Subspace MMSE Channel Estimation in Massive MU-MIMO System

Deng

Ohtsuki²

2020

IEEE Access

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Massive multiuser multiple-input multiple-output (massive MU-MIMO) technology is considered as a promising enabler to fulfill the rapid growth of traffic requirement for wireless mobile communications. The massive MU-MIMO system can achieve unlimited capacity when the base station (B-S) has accurate channel state information (CSI). In time-division-duplex (TDD) mode, the BS estimates CSI by receiving pilot signals sent from user terminals (UEs). However, because of using non-orthogonal pilots, pilot contamination happens to degrade the quality of the CSI estimation. To deal with pilot contamination problem, a low-complexity subspace minimum mean square error (MMSE) estimation method is proposed in this paper. Specifically, our approach operates the MMSE estimation in a low-dimensional subspace to avoid large matrix manipulation. Meanwhile, subspace projection helps to discriminate the desired signal and interfering signals in the power domain. Interference analysis shows the MMSE estimation can achieve interference-free estimation even in a low-dimensional subspace with a large number of BS antennas, and non-overlapping angles of arrival (AoAs) between desired and interfering UEs. Furthermore, thanks to the low-rank property of the channel covariance matrix in massive MU-MIMO systems, a two-step covariance matrix subspace projection method is proposed for further computational complexity reduction. The complexity analysis and simulation results indicate that our proposed approach has better channel estimation accuracy with lower complexity than the conventional MMSE estimation when the number of BS antennas is large. INDEX TERMS Massive MU-MIMO, channel estimation, pilot contamination, MMSE, subspace projection.

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A Rank-Revealing Method with Updating, Downdating, and Applications. Part II

Cited by 16 publications

References 19 publications

Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices

Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices

On the finite rank and finite-dimensional representation of bounded semi-infinite Hankel operators

Low-Complexity Subspace MMSE Channel Estimation in Massive MU-MIMO System

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