A Trace Minimization Algorithm for the Generalized Eigenvalue Problem

Sameh, Ahmed H.; Wisniewski, J.A.

doi:10.1137/0719089

Cited by 123 publications

(75 citation statements)

References 25 publications

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“…Theorem 1 (Sameh and Wisniewski [5]). Let A and B be n × n real symmetric matrices, with positive definite B, and let X be the set of all n×p matrices X for which…”

Section: Trace Minimization Eigensolvermentioning

confidence: 99%

See 1 more Smart Citation

A Parallel Implementation of the Trace Minimization Eigensolver

Romero

Román

2008

High Performance Computing for Computational Science - VECPAR 2008

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Abstract. In this paper we describe a parallel implementation of the trace minimization method for symmetric generalized eigenvalue problems proposed by Sameh and Wisniewski. The implementation includes several techniques proposed in a later article of Sameh, such as multishifting, preconditioning and adaptive inner solver termination, which accelerate the method and make it much more robust. A Davidson-type variant of the algorithm has been also considered. The different methods are analyzed in terms of sequential and parallel efficiency.Topics. Numerical algorithms for CS&E, parallel and distributed computing.

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“…Theorem 1 (Sameh and Wisniewski [5]). Let A and B be n × n real symmetric matrices, with positive definite B, and let X be the set of all n×p matrices X for which…”

Section: Trace Minimization Eigensolvermentioning

confidence: 99%

“…The trace minimization method for solving symmetric generalized eigenvalue problems was proposed by Sameh and Wisniewski [5], and developed further in [6]. The main idea of the method is to improve the update step of subspace iteration for generalized eigensystems as explained below.…”

Section: Trace Minimization Eigensolvermentioning

confidence: 99%

A Parallel Implementation of the Trace Minimization Eigensolver

Romero

Román

2008

High Performance Computing for Computational Science - VECPAR 2008

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“…Problem (14) is the classical minimum eigenvalue problem whose solution is given by the eigenvectors associated with the first smallest eigenvalues of ∆ [22]. Once the optimal M * is found, the optimal Ψ * can be recovered as…”

Section: Update Step For ψmentioning

confidence: 99%

Latent Space Sparse Subspace Clustering

Patel¹,

Nguyen²,

Vidal³

2013

2013 IEEE International Conference on Computer Vision

181

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We propose a novel algorithm called Latent Space Sparse Subspace Clustering for simultaneous dimensionality reduction and clustering of data lying in a union of subspaces. Specifically, we describe a method that learns the projection of data and finds the sparse coefficients in the low-dimensional latent space. Cluster labels are then assigned by applying spectral clustering to a similarity matrix built from these sparse coefficients. An efficient optimization method is proposed and its non-linear extensions based on the kernel methods are presented. One of the main advantages of our method is that it is computationally efficient as the sparse coefficients are found in the low-dimensional latent space. Various experiments show that the proposed method performs better than the competitive state-of-theart subspace clustering methods.

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“…Consider now the application of the correction equation (12) to the Domain Decomposition framework discussed above. Specifically, we consider the Newton approach discussed in Section 2.3 and we will apply one step of the Newton-Sylvester algorithm, i.e., Algorithm (2.1).…”

Section: Correction Equations and Domain Decompositionmentioning

confidence: 99%

“…In practice, this means that we need to solve the correction equation, i.e., the equation which updates the current approximate eigenvector, in a subspace that is orthogonal to the most current approximate eigenvectors. Several methods can be mentioned including the Trace Minimization method [12,11], the Davidson method [4,9] and the Jacobi-Davidson approach [14,13,16]. Most of these methods update an existing approximation by a step of Newton's method and this was illustrated in a number of papers, see, e.g., [8], and in [17].…”

Section: Introductionmentioning

confidence: 99%