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Minimization and parameter estimation for seminorm regularization models with Idivergence constraints
, 2012
"... In this papers we analyze the minimization of seminorms ‖L · ‖ on R n under the constraint of a bounded Idivergence D(b,H·) for rather general linear operators H and L. The Idivergence is also known as KullbackLeibler divergence and appears in many models in imaging science, in particular when d ..."
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In this papers we analyze the minimization of seminorms ‖L · ‖ on R n under the constraint of a bounded Idivergence D(b,H·) for rather general linear operators H and L. The Idivergence is also known as KullbackLeibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. We prove relations between the the parameters of Idivergence constrained and penalized problems without assuming the uniqueness of their minimizers. To solve the Idivergence constrained problem we apply firstorder primaldual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an Idivergence constrained least squares problem which can be solved based on Morosov’s discrepancy principle by a Newton method. Interestingly, the algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the corresponding penalized problem has the same solution as our constrained one. We demonstrate the performance of various algorithms for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise. 1
REGULARIZATION PARAMETER ESTIMATION FOR LARGE SCALE TIKHONOV REGULARIZATION USING A PRIORI INFORMATION
"... Abstract. This paper is concerned with estimating the solutions of numerically illposed least squares problems through Tikhonov regularization. Given a priori estimates on the covariance structure of errors in the measurement data b, and a suitable statisticallychosen σ, the Tikhonov regularized l ..."
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Abstract. This paper is concerned with estimating the solutions of numerically illposed least squares problems through Tikhonov regularization. Given a priori estimates on the covariance structure of errors in the measurement data b, and a suitable statisticallychosen σ, the Tikhonov regularized least squares functional J(σ) = ‖Ax − b ‖ 2 W b + 1/σ 2 ‖D(x − x0) ‖ 2 2, evaluated at its minimizer x(σ), approximately follows a χ2 distribution with ˜m degrees of freedom. Here ˜m = m + p − n for A ∈ Rm×n, D ∈ Rp×n, matrix Wb is the inverse covariance matrix of the mean 0 normally distributed measurement errors e in b, and x0 is an estimate of the mean value of x. Using the generalized singular value decomposition of the matrix pair [W 1/2 AD], σ can then be found such b that the resulting J follows this χ2 distribution, Mead and Renaut (2008). Because the algorithm explicitly relies on the direct solution of the problem obtained using the generalized singular value decomposition it is not practical for large scale problems. Here the approach is extended for large scale problems through the use of the Newton iteration in combination with a GolubKahan iterative bidiagonalization of the regularized problem. The algorithm is also extended for cases in which x0 is not available, but instead a set of measurement data provides an estimate of the
PORTFOLIO SELECTION USING TIKHONOV FILTERING TO ESTIMATE THE COVARIANCE MATRIX
"... Abstract. Markowitz’s portfolio selection problem chooses weights for stocks in a portfolio based on a covariance matrix of stock returns. Our study proposes to reduce noise in the estimated covariance matrix using a Tikhonov filter function. In addition, we propose a new strategy to resolve the ran ..."
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Abstract. Markowitz’s portfolio selection problem chooses weights for stocks in a portfolio based on a covariance matrix of stock returns. Our study proposes to reduce noise in the estimated covariance matrix using a Tikhonov filter function. In addition, we propose a new strategy to resolve the rank deficiency of the covariance matrix, and a method to choose a Tikhonov parameter which determines a filtering intensity. We put the previous estimators into a common framework and compare their filtering functions for eigenvalues of the correlation matrix. Experiments using the daily return data of the most frequently traded stocks in NYSE, AMEX, and NASDAQ show that Tikhonov filtering estimates the covariance matrix better than methods of Sharpe who applies a marketindex model, Ledoit et al. who shrink the sample covariance matrix to the marketindex covariance matrix, Elton and Gruber, who suggest truncating the smallest eigenvalues, Bengtsson and Holst, who decrease small eigenvalues at a single rate, and Plerou et al. and Laloux et al., who use a random matrix approach. Key words. Tikhonov regularization, covariance matrix estimate, Markowitz portfolio selection, ridge regression 1. Introduction. A
On the Use of Arnoldi and GolubKahan Bases to Solve Nonsymmetric IllPosed Inverse Problems
, 2015
"... Iterative Krylov subspace methods have proven to be efficient tools for solving linear systems of equations. In the context of illposed inverse problems, they tend to exhibit semiconvergence behavior making it difficult detect “inverted noise ” and stop iterations before solutions become contamina ..."
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Iterative Krylov subspace methods have proven to be efficient tools for solving linear systems of equations. In the context of illposed inverse problems, they tend to exhibit semiconvergence behavior making it difficult detect “inverted noise ” and stop iterations before solutions become contaminated. Regularization methods such as spectral filtering methods use the singular value decomposition (SVD) and are effective at filtering inverted noise from solutions, but are computationally prohibitive on large problems. Hybrid methods apply regularization techniques to the smaller “projected problem ” that is inherent to iterative Krylov methods at each iteration, thereby overcoming the semiconvergence behavior. Commonly, the GolubKahan bidiagonalization is used to construct a set of orthonormal basis vectors that span the Krylov subspaces from which solutions will be chosen, but seeking a solution in the orthonormal basis generated by the Arnoldi process (which is fundamental to the popular iterative method GMRES) has been of renewed interest recently. We discuss some of the positive and negative aspects of each process and use example problems to examine some qualities of the bases they produce. Computing optimal solutions in a given basis gives some insight into the performance of the corresponding iterative methods and
Acoustic imaging in enclosed spaces
, 2015
"... To cite this version: Antonio Pereira. Acoustic imaging in enclosed spaces. Acoustics [physics.classph]. INSA de ..."
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To cite this version: Antonio Pereira. Acoustic imaging in enclosed spaces. Acoustics [physics.classph]. INSA de
Insa: R. GOURDON
"... THÈSE Acoustic imaging in enclosed spaces présentée devant l’Institut National des Sciences Appliquées de Lyon par ..."
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THÈSE Acoustic imaging in enclosed spaces présentée devant l’Institut National des Sciences Appliquées de Lyon par
Reconstruction of the primordial power spectrum of curvature perturbations using multiple data sets
 PREPARED FOR SUBMISSION TO JCAP
, 2013
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ARTICLE IN PRESS Computational Statistics and Data Analysis ( ) – Contents lists available at ScienceDirect Computational Statistics and Data Analysis
"... journal homepage: www.elsevier.com/locate/csda Regularization parameter estimation for largescale Tikhonov ..."
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journal homepage: www.elsevier.com/locate/csda Regularization parameter estimation for largescale Tikhonov
estimation inverse problem for solid oxide
"... and error analysis of the polarization ..."
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