Sparse quasi-Newton updates with positive definite matrix completion

Yamashita, Nobuo

doi:10.1007/s10107-007-0137-1

Cited by 25 publications

(57 citation statements)

References 19 publications

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“…In addition, it is shown in [19] that the update does not suffer from Sorensen's example. Therefore, by relaxing the secant equation, the MCQN update is easy to be implemented and is well posed.…”

Section: ð1:9þmentioning

confidence: 97%

“…Numerical experiments in [19] show that the MCQN update with BFGS obviously performs better than that with DFP. However, local and superlinear convergence results are only established for the latter.…”

Section: ð1:9þmentioning

confidence: 99%

“…Toint [18] and Fletcher [7] Further, Yamashita showed that, if the sparsity of the Hessian is of such pattern that there is no fill-in in its Cholesky factorization, or equivalently, the graph induced by the Hessian is chordal (see [19] for details), the problem (1. …”

Section: ð1:5þmentioning

confidence: 99%

“…The above problem can then be easily solved by analyzing the clique tree of the graph induced by the Hessian (see [19] for details). In addition, it is shown in [19] that the update does not suffer from Sorensen's example.…”

Section: ð1:9þmentioning

confidence: 99%

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Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion

Dai

Yamashita

2014

J. Oper. Res. Soc. China

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Based on the idea of maximum determinant positive definite matrix completion, Yamashita (Math Prog 115(1):1-30, 2008) proposed a new sparse quasi-Newton update, called MCQN, for unconstrained optimization problems with sparse Hessian structures. In exchange of the relaxation of the secant equation, the MCQN update avoids solving difficult subproblems and overcomes the ill-conditioning of approximate Hessian matrices. However, local and superlinear convergence results were only established for the MCQN update with the DFP method. In this paper, we extend the convergence result to the MCQN update with the whole Broyden's convex family. Numerical results are also reported, which suggest some efficient ways of choosing the parameter in the MCQN update the Broyden's family.

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Section: ð1:9þmentioning

confidence: 97%

Section: ð1:9þmentioning

confidence: 99%

Section: ð1:5þmentioning

confidence: 99%

Section: ð1:9þmentioning

confidence: 99%

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Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion

Dai

Yamashita

2014

J. Oper. Res. Soc. China

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“…The chordal graph theory provides rich tools, in not only statistics but also many fields of mathematical science. In particular, in numerical analysis, this is a major tool used to conduct large-scale matrix computation (e.g., Rose 1971;Yamashita 2008). We also apply the chordal graph theory to retrieve the Markov structure to reduce the computational time using the recursive summation/integration technique.…”

Section: Introductionmentioning

confidence: 99%

Multiplicity adjustment for temporal and spatial scan statistics using Markov property

Kuriki

Takahashi

Hara

2018

Jpn J Stat Data Sci

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We discuss computation of the conditional expectation in a multinomial distribution. The problem is motivated by the evaluation of the multiplicity-adjusted p-value of scan statistics in spatial epidemiology. We propose a recursive summation/integration technique using the Markov property, which is extracted from a chordal graph defined by temporal and spatial structures. This methodology can be applied to a class of distributions, including the Poisson distribution (that is, the conditional distribution is the multinomial). To illustrate the approach, we present the real data analyses for detecting temporal and spatial clustering.

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On Listing, Sampling, and Counting the Chordal Graphs with Edge Constraints

Kijima

Kiyomi

Okamoto

et al.

Lecture Notes in Computer Science

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We discuss the problems to list, sample, and count the chordal graphs with edge constraints. The edge constraints are given as a pair of graphs one of which contains the other and one of which is chordal, and the objects we look at are the chordal graphs contained in one and containing the other. This setting is a natural generalization of chordal completions and deletions. For the listing problem, we give an efficient algorithm running in amortized polynomial time per output with polynomial space. For the sampling problem, we give an instance for which a natural Markov chain suffers from an exponential mixing time. For the counting problem, we show some #P-completeness results. These results provide a unified viewpoint from algorithms theory to problems arising from various areas such as statistics, data mining, and numerical computation.

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Sparse quasi-Newton updates with positive definite matrix completion

Cited by 25 publications

References 19 publications

Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion

Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion

Multiplicity adjustment for temporal and spatial scan statistics using Markov property

On Listing, Sampling, and Counting the Chordal Graphs with Edge Constraints

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