Parameter Estimation in the Presence of Bounded Data Uncertainties

Chandrasekaran, S.; Golub, Gene H.; Gu, Ming; Sayed, Ali H.

doi:10.1137/s0895479896301674

Cited by 138 publications

(104 citation statements)

References 8 publications

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“…11], [19][20][21]. These were followed by many results for specific problem classes or applications; see, e.g., the survey [22]; examples include robust linear programs [2,23,24], robust least-squares [25,26], robust quadratically constrained programs [27], robust semidefinite programs [28], robust conic programming [29], robust discrete optimization [30]. Work focused on specific applications includes robust control [31,32], robust portfolio optimization [33][34][35][36], robust beamforming [37][38][39], robust machine learning [40], and many others.…”

Section: Worst-case Robust Optimizationmentioning

confidence: 99%

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Cutting-set methods for robust convex optimization with pessimizing oracles

Mutapcic

Boyd

2009

Optimization Methods and Software

128

170

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We consider a general worst-case robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worst-case analysis. With exact worst-case analysis, the method is shown to converge to a robust optimal point. With approximate worst-case analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worst-case analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps; using warm-start techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples.

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Section: Worst-case Robust Optimizationmentioning

confidence: 99%

“…We will use the basic CSM to solve (26), with an exact pessimization oracle we describe below. For work on other robust beamforming problems (but not including this uncertainty model), see [37-39, 90, 91].…”

Section: Robust Beamforming With Uncertain Locationsmentioning

confidence: 99%

Cutting-set methods for robust convex optimization with pessimizing oracles

Mutapcic

Boyd

2009

Optimization Methods and Software

128

170

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“…The solution of (4) is shown to be exactly the regularized LS solution (2). Using the singular value decomposition (SVD) A = U ΛV T , the parameter γ is obtained by solving the secular equation [8] …”

Section: The Basic Bdu Approachmentioning

confidence: 99%

“…More specifically, the emphasis is on the case where N is not much larger than M , which is the case where the ordinary LS estimator suffers most. We adopt a bounded data uncertainty (BDU) approach [8]. In the BDU model, the measurement matrix is not precisely known due to an unknown perturbation.…”

Section: Introductionmentioning

confidence: 99%

Improved linear least squares estimation using bounded data uncertainty

Ballal

Al-Naffouri

2015

2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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This paper addresses the problem of linear least squares (LS) estimation of a vector x from linearly related observations. In spite of being unbiased, the original LS estimator suffers from high mean squared error, especially at low signal-to-noise ratios. The mean squared error (MSE) of the LS estimator can be improved by introducing some form of regularization based on certain constraints. We propose an improved LS (ILS) estimator that approximately minimizes the MSE, without imposing any constraints. To achieve this, we allow for perturbation in the measurement matrix. Then we utilize a bounded data uncertainty (BDU) framework to derive a simple iterative procedure to estimate the regularization parameter. Numerical results demonstrate that the proposed BDU-ILS estimator is superior to the original LS estimator, and it converges to the best linear estimator, the linear-minimum-mean-squared error estimator (LMMSE), when the elements of x are statistically white.

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“…Similar ideas of using the worst-case robust optimization have been successfully applied to related signal processing problems such as: robust filtering [18], [19], robust parameter estimation [20], [21], robust matched filtering [22], [23], robust minimum variance beamforming [24]- [28].…”

Section: Related Workmentioning

confidence: 99%

Robust Chebyshev FIR Equalization

Mutapcic

Kim

Boyd

2007

IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference

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Abstract-In Chebyshev finite-impulse response (FIR) equalization, we design an FIR filter that minimizes the Chebyshev equalization error, i.e., the maximum absolute deviation between the equalized and the desired frequency response functions, assuming the unequalized response function is known exactly. In robust Chebyshev FIR equalization, we take into account uncertainty in the unequalized response function, described as a set of possible values for the unequalized response at each frequency, by designing an FIR filter that minimizes worst-case Chebyshev equalization error over all possible unequalized response functions. When the uncertainty in unequalized response function is described by a complex uncertainty ellipsoid, at each frequency, we show that the robust Chebyshev FIR equalization design problem can be formulated as a semidefinite program (SDP), and therefore efficiently (and globally) solved. When the uncertainty is given by a complex disk, the design problem can be formulated as a second-order cone program (SOCP), which can be solved almost as fast as the nominal Chebyshev equalization problem (ignoring uncertainty). The robust equalizer design method is demonstrated with a numerical example.

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Parameter Estimation in the Presence of Bounded Data Uncertainties

Cited by 138 publications

References 8 publications

Cutting-set methods for robust convex optimization with pessimizing oracles

Cutting-set methods for robust convex optimization with pessimizing oracles

Improved linear least squares estimation using bounded data uncertainty

Robust Chebyshev FIR Equalization

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