Superfast and Stable Structured Solvers for Toeplitz Least Squares via Randomized Sampling

Xi, Yuanzhe; Xia, Jianlin; Cauley, Stephen F.; Balakrishnan, V.

doi:10.1137/120895755

Cited by 64 publications

(82 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that for arbitrary k-space trajectories, the Gram matrix , or the coefficient matrix of (17), has a block Toeplitz structure [25]. A number of off-the-shelf numerical solvers taking advantage of this structure enables very efficient solution of (17), including the pre-conditioned CG solvers (e.g., [26], [27]) and hierarchically semiseparable (HSS) solvers (e.g., [28], [29]). (5) is a non-convex optimization problem, the solution of the algorithm generally depends on the initialization.…”

Section: Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Maximum likelihood reconstruction for magnetic resonance fingerprinting

Zhao

Lam

Bilgic̦

et al. 2015

2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI)

View full text Add to dashboard Cite

This paper introduces a statistical estimation framework for magnetic resonance (MR) fingerprinting, a recently proposed quantitative imaging paradigm. Within this framework, we present a maximum likelihood (ML) formalism to estimate multiple parameter maps directly from highly undersampled, noisy k-space data. A novel algorithm, based on variable splitting, the alternating direction method of multipliers, and the variable projection method, is developed to solve the resulting optimization problem. Representative results from both simulations and in vivo experiments demonstrate that the proposed approach yields significantly improved accuracy in parameter estimation, compared to the conventional MR fingerprinting reconstruction. Moreover, Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. HHS Public Access Author Manuscript Author ManuscriptAuthor ManuscriptAuthor Manuscript the proposed framework provides new theoretical insights into the conventional approach. We show analytically that the conventional approach is an approximation to the ML reconstruction; more precisely, it is exactly equivalent to the first iteration of the proposed algorithm for the ML reconstruction, provided that a gridding reconstruction is used as an initialization.

show abstract

Section: Algorithmmentioning

confidence: 99%

“…Solving each subproblem only involves inverting a diagonal matrix, which can be done in (N) operations. Regarding the linear least-squares problem in (10), it is decoupled with respect to each time frame and each coil, and each subproblem can be solved by the HSS solver in (N log 2 N) operations [28].…”

Section: )mentioning

confidence: 99%

Maximum likelihood reconstruction for magnetic resonance fingerprinting

Zhao

Lam

Bilgic̦

et al. 2015

2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI)

View full text Add to dashboard Cite

show abstract

“…Despite the large number of references available on fast algorithms for computations with low-rank structured matrices, there exist very few references on the corresponding a priori rounding error analyses [1,2,4,14,18,19]. Some reasons of this might be that such fast algorithms are frequently complicated and that some of them are potentially unstable, although work well in practice most of the times.…”

Section: Introductionmentioning

confidence: 99%

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

Dopico

Pomés

2016

Numer Algor

View full text Add to dashboard Cite

Low-rank structured matrices have attracted much attention in the last decades, since they arise in many applications and all share the fundamental property that can be represented by O(n) parameters, where n × n is the size of the matrix. This property has allowed the development of fast algorithms for solving numerically many problems involving low-rank structured matrices by performing operations on the parameters describing the matrices, instead of directly on the matrix entries. Among these problems, the solution of linear systems of equations is probably the most basic and relevant one. Therefore, it is important to measure, via structured computable condition numbers, the relative sensitivity of the solutions of linear systems with low-rank structured coefficient matrices with respect to relative perturbations of the parameters representing such matrices, since this sensitivity determines the maximum accuracy attainable by fast algorithms and allows us to decide which set of parameters is the most convenient from the point of view of accuracy. To develop and analyze such condition numbers is the main goal of this paper. To this purpose, a general expression is obtained for the condition number of the solution of a linear system of equations whose coefficient matrix is any differentiable function of a vector of parameters with respect to perturbations of such parameters. Since there are many different classes of low-rank structured matrices and many different types of parameters Partially supported by Ministerio de Economía y Competitividad of Spain through grants describing them, it is not possible to cover all of them in a single work. Therefore, the general expression of the condition number is particularized to the important case of {1, 1}-quasiseparable matrices and to the quasiseparable and the Givens-vector representations, in order to obtain explicit expressions of the corresponding two condition numbers that can be estimated in O(n) operations. In addition, detailed theoretical and numerical comparisons of these two condition numbers between themselves, and with respect to unstructured condition numbers, are provided, which show that there are situations in which the unstructured condition number is much larger than the structured ones, but that the opposite never happens. The approach presented in this manuscript can be generalized to other classes of low-rank structured matrices and parameterizations.

show abstract

“…Once an HSS approximation to A is constructed, the well-established fast HSS factorization and solution algorithms in [7,34,37] can be applied.…”

Section: Randomized Hss Construction: Adaptive and Matrix-free Schemesmentioning

confidence: 99%

“…[34] that such type of HSS factorizations usually have much better stability than standard dense factorizations (for the same matrix) due to the hierarchical structure and the use of orthogonal off-diagonal operations.…”

Section: Or (48)mentioning

confidence: 99%

A Fast Randomized Eigensolver with Structured LDL Factorization Update

Xi¹,

Xia²,

Chan³

2014

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper, we propose a structured bisection method with adaptive randomized sampling for finding selected or all of the eigenvalues of certain real symmetric matrices A. For A with a low-rank property, we construct a hierarchically semiseparable (HSS) approximation and show how to quickly evaluate and update its inertia in the bisection method. Unlike some existing randomized HSS constructions, the methods here do not require the knowledge of the off-diagonal (numerical) ranks in advance. Moreover, for A with a weak rank property or slowly decaying offdiagonal singular values, we show an idea of aggressive low-rank inertia evaluation, which means that a compact HSS approximation can preserve the inertia for certain shifts. This is analytically justified for a special case, and numerically shown for more general ones. A generalized LDL factorization of the HSS approximation is then designed for the fast evaluation of the inertia. A significant advantage over standard LDL factorizations is that the HSS LDL factorization (and thus the inertia) of A − sI can be quickly updated with multiple shifts s in bisection. The factorization with each new shift can reuse about 60% of the work. As an important application, the structured eigensolver can be applied to symmetric Toeplitz matrices, and the cost to find one eigenvalue is nearly linear in the order of the matrix. The numerical examples demonstrate the efficiency and the accuracy of our methods, especially the benefit of low-rank inertia evaluations. The ideas and methods can be potentially adapted to other HSS computations where shifts are involved and to more problems without a significant low-rank property.

show abstract

Superfast and Stable Structured Solvers for Toeplitz Least Squares via Randomized Sampling

Cited by 64 publications

References 34 publications

Maximum likelihood reconstruction for magnetic resonance fingerprinting

Maximum likelihood reconstruction for magnetic resonance fingerprinting

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

A Fast Randomized Eigensolver with Structured LDL Factorization Update

Contact Info

Product

Resources

About