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218
EM Algorithms for PCA and SPCA
 in Advances in Neural Information Processing Systems
, 1998
"... I present an expectationmaximization (EM) algorithm for principal component analysis (PCA). The algorithm allows a few eigenvectors and eigenvalues to be extracted from large collections of high dimensional data. It is computationally very efficient in space and time. It also naturally accommodates ..."
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Cited by 146 (1 self)
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I present an expectationmaximization (EM) algorithm for principal component analysis (PCA). The algorithm allows a few eigenvectors and eigenvalues to be extracted from large collections of high dimensional data. It is computationally very efficient in space and time. It also naturally accommodates missing information. I also introduce a new variant of PCA called sensible principal component analysis (SPCA) which defines a proper density model in the data space. Learning for SPCA is also done with an EM algorithm. I report results on synthetic and real data showing that these EM algorithms correctly and efficiently find the leading eigenvectors of the covariance of datasets in a few iterations using up to hundreds of thousands of datapoints in thousands of dimensions.
DyRT: Dynamic Response Textures for Real Time Deformation Simulation with Graphics Hardware
, 2002
"... In this paper we describe how to simulate geometrically complex, interactive, physicallybased, volumetric, dynamic deformation models with negligible main CPU costs. This is achieved using a Dynamic Response Texture, or DyRT, that can be mapped onto any conventional animation as an optional renderi ..."
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Cited by 96 (13 self)
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In this paper we describe how to simulate geometrically complex, interactive, physicallybased, volumetric, dynamic deformation models with negligible main CPU costs. This is achieved using a Dynamic Response Texture, or DyRT, that can be mapped onto any conventional animation as an optional rendering stage using commodity graphics hardware. The DyRT simulation process employs precomputed modal vibration models excited by rigid body motions. We present several examples, with an emphasis on bonebased character animation for interactive applications.
An ArnoldiSchur Algorithm for Large Eigenproblems
, 2000
"... Sorensen's iteratively restarted Arnoldi algorithm is one of the most successful and flexible methods for finding a few eigenpairs of a large matrix. However, the need to preserve structure of the Arnoldi decomposition, on which the algorithm is based, restricts the range of transformations ..."
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Cited by 64 (2 self)
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Sorensen's iteratively restarted Arnoldi algorithm is one of the most successful and flexible methods for finding a few eigenpairs of a large matrix. However, the need to preserve structure of the Arnoldi decomposition, on which the algorithm is based, restricts the range of transformations that can be performed on it. In consequence, it is difficult to deflate converged Ritz vectors from the decomposition. Moreover, the potential forward instability of the implicit QR algorithm can cause unwanted Ritz vectors to persist in the computation. In this paper we introduce a generalized Arnoldi decomposition that solves both problems in a natural and efficient manner.
A New MatrixFree Algorithm for the LargeScale TrustRegion Subproblem
, 1995
"... The trustregion subproblem arises frequently in linear algebra and optimization applications. Recently, matrixfree methods have been introduced to solve large scale trustregion subproblems. These methods only require a matrixvector product and do not rely on matrix factorizations [4, 7]. The ..."
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Cited by 63 (11 self)
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The trustregion subproblem arises frequently in linear algebra and optimization applications. Recently, matrixfree methods have been introduced to solve large scale trustregion subproblems. These methods only require a matrixvector product and do not rely on matrix factorizations [4, 7]. These approaches recast the trust region subproblem in terms of a parameterized eigenvalue problem and then adjust the parameter to find the optimal solution from the eigenvector corresponding to the smallest eigenvalue of the parameterized eigenvalue problem. This paper presents a new matrixfree algorithm for the largescale trustregion subproblem. The new algorithm improves upon the previous algorithms by introducing a unified iteration that naturally includes the so called hard case. The new iteration is shown to be superlinearly convergent in all cases. Computational results are presented to illustrate convergence properties and robustness of the method.
Knowledge mining with VxInsight: Discovery through interaction
 JOURNAL OF INTELLIGENT INFORMATION SYSTEMS
, 1998
"... The explosive growth in the availability of information is overwhelming traditional information management systems. Although individual pieces of information have become easy to find, the larger context in which they exist has become harder to track. These contextual questions are ideally suited to ..."
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Cited by 54 (5 self)
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The explosive growth in the availability of information is overwhelming traditional information management systems. Although individual pieces of information have become easy to find, the larger context in which they exist has become harder to track. These contextual questions are ideally suited to visualization since the humrex visual system is remarkably adept at interpreting large quantities of information, and at detecting patterns and anomalies. The challenge is to present the information in a manner that maximally leverages our v/sual skills. This paper discusses a set of properties that such a presentation should have, and describes the design and functionality of Vxlnsight, a visualization tool built to these principles.
The Sylvester equation and approximate balanced reduction
, 2002
"... The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection me ..."
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Cited by 53 (4 self)
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The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Pad via Lanczos) and rationa Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the cross gramtan of the system. The cross gramtan is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights.
ABLE: an adaptive block Lanczos method for nonHermitian eigenvalue problems
 SIAM Journal on Matrix Analysis and Applications
, 1999
"... Abstract. This work presents an adaptive block Lanczos method for largescale nonHermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the nonHermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) break ..."
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Cited by 49 (2 self)
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Abstract. This work presents an adaptive block Lanczos method for largescale nonHermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the nonHermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria are developed that exploit the semiquadratic convergence property of the method. Third, a wellknown technique from the Hermitian Lanczos algorithm is generalized to monitor the loss of biorthogonality and maintain semibiorthogonality among the computed Lanczos vectors. Each innovation is theoretically justified. Academic model problems and real application problems are solved to demonstrate the numerical behaviors of the method. Key words. method nonHermitian matrices, eigenvalue problem, spectral transformation, Lanczos AMS subject classifications. 65F15, 65F10 PII. S0895479897317806
A reliable and computationally efficient algorithm for imposing the saddle point property in dynamic models. Manuscript, Federal Reserve Board of Governors
"... linear saddle point models. The algorithm has proved useful in a wide array of applications including analyzing linear perfect foresight models, providing initial solutions and asymptotic constraints for nonlinear models. The algorithm solves linear problems with dozens of lags and leads and hundred ..."
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Cited by 38 (2 self)
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linear saddle point models. The algorithm has proved useful in a wide array of applications including analyzing linear perfect foresight models, providing initial solutions and asymptotic constraints for nonlinear models. The algorithm solves linear problems with dozens of lags and leads and hundreds of equations in seconds. The technique works well for both symbolic algebra and numerical computation. Although widely used at the Federal Reserve, few outside the central bank know about or have used the algorithm. This paper attempts to present the current algorithm in a more accessible format in the hope that economists outside the Federal Reserve may also nd it useful. In addition, over the years there have been undocumented changes in approach that have improved the eciency and reliability of algorithm. This paper describes the present state of development of this set of tools.
A trustregion approach to the regularization of largescale discrete forms of illposed problems
 SISC
, 2000
"... We consider largescale least squares problems where the coefficient matrix comes from the discretization of an operator in an illposed problem, and the righthand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effe ..."
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Cited by 30 (8 self)
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We consider largescale least squares problems where the coefficient matrix comes from the discretization of an operator in an illposed problem, and the righthand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution. We pose the regularization problem as a quadratically constrained least squares problem. This formulation is equivalent to Tikhonov regularization, and we note that it is also a special case of the trustregion subproblem from optimization. We analyze the trustregion subproblem in the regularization case, and we consider the nontrivial extensions of a recently developed method for general largescale subproblems that will allow us to handle this case. The method relies on matrixvector products only, has low and fixed storage requirements, and can handle the singularities arising in illposed problems. We present numerical results on test problems, on an
Implicitly restarted Arnoldi methods and eigenvalues of the discretized Navier Stokes equations.
 SIAM J. Matrix Anal. Appl
, 1997
"... We are concerned with finding a few eigenvalues of the large sparse nonsymmetric generalized eigenvalue problem Ax = Bx that arises in stability studies of incompressible fluid flow. The matrices have a block structure that is typical of mixed finiteelement discretizations for such problems. We ..."
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Cited by 30 (3 self)
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We are concerned with finding a few eigenvalues of the large sparse nonsymmetric generalized eigenvalue problem Ax = Bx that arises in stability studies of incompressible fluid flow. The matrices have a block structure that is typical of mixed finiteelement discretizations for such problems. We examine the use of shiftinvert and Cayley transformations in conjunction with the implicitly restarted Arnoldi method along with using a semiinner product induced by B and purification techniques. Numerical results are presented for some model problems arising from the ENTWIFE finiteelement package. Our conclusion is that, with careful implementation, implicitly restarted Arnoldi methods are reliable for linear stability analysis. AMS classification: Primary 65F15; Secondary 65F50 Key Words: eigenvalues, sparse nonsymmetric matrices, Arnoldi's method. 1 Introduction Mixed finiteelement discretizations of timedependent equations modelling incompressible fluid flow problems ty...