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198
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... . We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpret ..."
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Cited by 145 (12 self)
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. We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomialtime using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Leastsquares, uncertainty, robustness, secondorder cone...
A Framework for Robust Subspace Learning
 International Journal of Computer Vision
, 2003
"... Many computer vision, signal processing and statistical problems can be posed as problems of learning low dimensional linear or multilinear models. These models have been widely used for the representation of shape, appearance, motion, etc, in computer vision applications. ..."
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Cited by 96 (6 self)
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Many computer vision, signal processing and statistical problems can be posed as problems of learning low dimensional linear or multilinear models. These models have been widely used for the representation of shape, appearance, motion, etc, in computer vision applications.
Robust computation of optic flow in a multiscale differential framework
 International Journal of Computer Vision
, 1995
"... Abstract. We have developed a new algorithm for computing optical flow in a differential framework. The image sequence is first convolved with a set of linear, separable spatiotemporal filter kernels similar to those that have been used in other early vision problems such as texture and stereopsis. ..."
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Cited by 96 (2 self)
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Abstract. We have developed a new algorithm for computing optical flow in a differential framework. The image sequence is first convolved with a set of linear, separable spatiotemporal filter kernels similar to those that have been used in other early vision problems such as texture and stereopsis. The brightness constancy constraint can then be applied to each of the resulting images, giving us, in general, an overdetermined system of equations for the optical flow at each pixel. There are three principal sources of error: (a) stochastic error due to sensor noise (b) systematic errors in the presence of large displacements and (c) errors due to failure of the brightness constancy model. Our analysis of these errors leads us to develop an algorithm based on a robust version of total least squares. Each optical flow vector computed has an associated reliability measure which can be used in subsequent processing. The performance of the algorithm on the data set used by Barron et al. (IJCV 1994) compares favorably with other techniques. In addition to being separable, the filters used are also causal, incorporating only past time frames. The algorithm is fully parallel and has been implemented on a multiple processor machine. 1
Computing Optical Flow with Physical Models of Brightness Variation
"... This paper exploits physical models of timevarying brightness in image sequences to estimate optical flow and physical parameters of the scene. Previous approaches handled violations of brightness constancy with the use of robust statistics or with generalized brightness constancy constraints that ..."
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Cited by 85 (1 self)
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This paper exploits physical models of timevarying brightness in image sequences to estimate optical flow and physical parameters of the scene. Previous approaches handled violations of brightness constancy with the use of robust statistics or with generalized brightness constancy constraints that allow generic types of contrast and illumination changes. Here, we consider models of brightness variation that have timedependent physical causes, namely, changing surface orientation with respect to a directional illuminant, motion of the illuminant, and physical models of heat transport in infrared images. We simultaneously estimate the optical flow and the relevant physical parameters. The estimation problem is formulated using total least squares (TLS), with confidence bounds on the parameters.
SLICOT  A Subroutine Library in Systems and Control Theory
 Applied and Computational Control, Signals, and Circuits
, 1997
"... This article describes the subroutine library SLICOT that provides Fortran 77 implementations of numerical algorithms for computations in systems and control theory. Around a nucleus of basic numerical linear algebra subroutines, this library builds methods for the design and analysis of linear cont ..."
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Cited by 74 (53 self)
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This article describes the subroutine library SLICOT that provides Fortran 77 implementations of numerical algorithms for computations in systems and control theory. Around a nucleus of basic numerical linear algebra subroutines, this library builds methods for the design and analysis of linear control systems. A brief history of the library is given together with a description of the current version of the library and the ongoing activities to complete and improve the library in several aspects. 1 Introduction Systems and control theory are disciplines widely used to describe, control, and optimize industrial and economical processes. There is now a huge amount of theoretical results available which has lead to a variety of methods and algorithms used throughout industry and academia. Although based on theoretical results, these methods often fail when applied to reallife problems, which often tend to be illposed or of high dimensions. This failing is frequently due to the lack of...
TIKHONOV REGULARIZATION AND TOTAL LEAST SQUARES
 SIAM J. MATRIX ANAL. APPL
, 1999
"... Discretizations of inverse problems lead to systems of linear equations with a highly illconditioned coefficient matrix, and in order to compute stable solutions to these systems it is necessary to apply regularization methods. We show how Tikhonov’s regularization method, which in its original for ..."
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Cited by 56 (2 self)
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Discretizations of inverse problems lead to systems of linear equations with a highly illconditioned coefficient matrix, and in order to compute stable solutions to these systems it is necessary to apply regularization methods. We show how Tikhonov’s regularization method, which in its original formulation involves a least squares problem, can be recast in a total least squares formulation suited for problems in which both the coefficient matrix and the righthand side are known only approximately. We analyze the regularizing properties of this method and demonstrate by a numerical example that, in certain cases with large perturbations, the new method is superior to standard regularization methods.
Analysis of incomplete climate data: Estimation of mean values and covariance matrices and imputation of missing values
, 2001
"... Estimating the mean and the covariance matrix of an incomplete dataset and filling in missing values with imputed values is generally a nonlinear problem, which must be solved iteratively. The expectation maximization (EM) algorithm for Gaussian data, an iterative method both for the estimation of m ..."
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Cited by 55 (3 self)
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Estimating the mean and the covariance matrix of an incomplete dataset and filling in missing values with imputed values is generally a nonlinear problem, which must be solved iteratively. The expectation maximization (EM) algorithm for Gaussian data, an iterative method both for the estimation of mean values and covariance matrices from incomplete datasets and for the imputation of missing values, is taken as the point of departure for the development of a regularized EM algorithm. In contrast to the conventional EM algorithm, the regularized EM algorithm is applicable to sets of climate data, in which the number of variables typically exceeds the sample size. The regularized EM algorithm is based on iterated analyses of linear regressions of variables with missing values on variables with available values, with regression coefficients estimated by ridge regression, a regularized regression method in which a continuous regularization parameter controls the filtering of the noise in the data. The regularization parameter is determined by generalized crossvalidation, such as to minimize, approximately, the expected mean squared error of the imputed values. The regularized EM algorithm can estimate, and exploit for the imputation of missing values, both synchronic and diachronic covariance matrices, which may contain information on spatial covariability, stationary temporal covariability, or cyclostationary temporal covariability. A test of the regularized EM algorithm with simulated surface temperature data demonstrates that the algorithm is applicable to typical sets of climate data and that it leads to more accurate estimates of the missing values than a conventional noniterative imputation technique.
Robust meansquared error estimation in the presence of model uncertainties
 IEEE Trans. on Signal Processing
, 2005
"... Abstract—We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties, where the vector x is known to satisfy a weighted norm constraint. We first assume that the model is known exactly and seek the linear estimator that minimizes the wor ..."
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Cited by 52 (37 self)
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Abstract—We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties, where the vector x is known to satisfy a weighted norm constraint. We first assume that the model is known exactly and seek the linear estimator that minimizes the worstcase meansquared error (MSE) across all possible values of x. We show that for an arbitrary choice of weighting, the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved very efficiently. We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke, with a specific choice of shrinkage factor that explicitly takes the prior information into account. Next, we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worstcase MSE across all possible values of x and all possible values of the model matrix. As we show, the robust minimax MSE estimator can also be formulated as a solution to an SDP. Finally, we demonstrate through several examples that the minimax MSE estimator can significantly increase the performance over the conventional leastsquares estimator, and when the model matrix is subject to uncertainties, the robust minimax MSE estimator can lead to a considerable improvement in performance over the minimax MSE estimator. Index Terms—Data uncertainty, linear estimation, mean squared error estimation, minimax estimation, robust estimation. I.
Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 50 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.