The Experts below are selected from a list of 294 Experts worldwide ranked by ideXlab platform
Mihailo R. Jovanovic - One of the best experts on this subject based on the ideXlab platform.
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Least Squares Approximation of structured covariances
IEEE Transactions on Automatic Control, 2009Co-Authors: Fu Lin, Mihailo R. JovanovicAbstract:State covariances of linear systems satisfy certain constraints imposed by the underlying dynamics. These constraints dictate a particular structure of state covariances. However, sample covariances almost always fail to have the required structure. The renewed interest in using state covariances for estimating the power spectra of inputs gives rise to the Approximation problem. In this note, the structured covariance Least-Squares problem is formulated and the Lyapunov-type matricial linear constraint is converted into an equivalent set of trace constraints. Efficient unconstrained maximization methods capable of solving the corresponding dual problem are developed.
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ACC - On the Least-Squares Approximation of structured covariances
2007 American Control Conference, 2007Co-Authors: Fu Lin, Mihailo R. JovanovicAbstract:State covariances of the linear systems satisfy certain constraints imposed by the underlying dynamics. These constraints dictate a particular structure of state covariances. On the other hand, sample covariances (e.g., obtained in experiments) almost always fail to have the required structure. In view of this, it is of interest to approximate sample covariances by positive semi-definite matrices of the required structure. The structured covariance Least-Squares Approximation problem is formulated and the Lyapunov-type matrical linear constraint is converted into an equivalent set of trace constraints. Efficient quasi Newton and generalized Newton methods capable of solving the corresponding unconstrained dual problems with the large number of variables are developed.
Fu Lin - One of the best experts on this subject based on the ideXlab platform.
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Least Squares Approximation of structured covariances
IEEE Transactions on Automatic Control, 2009Co-Authors: Fu Lin, Mihailo R. JovanovicAbstract:State covariances of linear systems satisfy certain constraints imposed by the underlying dynamics. These constraints dictate a particular structure of state covariances. However, sample covariances almost always fail to have the required structure. The renewed interest in using state covariances for estimating the power spectra of inputs gives rise to the Approximation problem. In this note, the structured covariance Least-Squares problem is formulated and the Lyapunov-type matricial linear constraint is converted into an equivalent set of trace constraints. Efficient unconstrained maximization methods capable of solving the corresponding dual problem are developed.
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ACC - On the Least-Squares Approximation of structured covariances
2007 American Control Conference, 2007Co-Authors: Fu Lin, Mihailo R. JovanovicAbstract:State covariances of the linear systems satisfy certain constraints imposed by the underlying dynamics. These constraints dictate a particular structure of state covariances. On the other hand, sample covariances (e.g., obtained in experiments) almost always fail to have the required structure. In view of this, it is of interest to approximate sample covariances by positive semi-definite matrices of the required structure. The structured covariance Least-Squares Approximation problem is formulated and the Lyapunov-type matrical linear constraint is converted into an equivalent set of trace constraints. Efficient quasi Newton and generalized Newton methods capable of solving the corresponding unconstrained dual problems with the large number of variables are developed.
Gregory E. Fasshauer - One of the best experts on this subject based on the ideXlab platform.
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preconditioning of radial basis function interpolation systems via accelerated iterated approximate moving Least Squares Approximation
2009Co-Authors: Gregory E. Fasshauer, Jack G. ZhangAbstract:The standard approach to the solution of the radial basis function interpo- lation problem has been recognized as an ill-conditioned problem for many years. This is especially true when infinitely smooth basic functions such as multiquadrics or Gaussians are used with extreme values of their associated shape parameters. Various approaches have been described to deal with this phenomenon. These tech- niques include applying specialized preconditioners to the system matrix, changing the basis of the Approximation space or using techniques from complex analysis. In this paper we present a preconditioning technique based on residual iteration of an approximate moving Least Squares quasi-interpolant that can be interpreted as a change of basis. In the limit our algorithm will produce the perfectly conditioned cardinal basis of the underlying radial basis function Approximation space. Although our method is motivated by radial basis function interpolation problems, it can also be adapted for similar problems when the solution of a linear system is involved such as collocation methods for solving differential equations.
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Toward approximate moving Least Squares Approximation with irregularly spaced centers
Computer Methods in Applied Mechanics and Engineering, 2004Co-Authors: Gregory E. FasshauerAbstract:Abstract By combining the well-known moving Least Squares Approximation method and the theory of approximate Approximations due to Maz’ya and Schmidt we are able to present an approximate moving Least Squares method which inherits the simplicity of Shepard’s method along with the accuracy of higher-order moving Least Squares Approximations. In this paper we focus our interest on practical implementations for irregularly spaced data sites. The two schemes described here along with some first numerical experiments are to be viewed as exploratory work only. These schemes apply to centers that are obtained from gridded centers via a smooth parametrization. Further work to find a robust numerical scheme applicable to arbitrary scattered data is needed.
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Approximate Moving Least-Squares Approximation with Compactly Supported Radial Weights
Lecture Notes in Computational Science and Engineering, 2003Co-Authors: Gregory E. FasshauerAbstract:We use Maz’ya and Schmidt’s theory of approximate Approximation to devise a fast and accurate approximate moving Least-Squares Approximation method which does not require the solution of any linear systems. Since we use compactly supported weight functions, the remaining summation is also efficient. We compare our new algorithm with three other Approximation methods based on compactly supported radial functions: multilevel interpolation, the standard moving Least-Squares Approximation method, and a multilevel moving Least-Squares algorithm. A multilevel approximate moving Least-Squares Approximation algorithm is also included.
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approximate moving Least Squares Approximation a fast and accurate multivariate Approximation method
2002Co-Authors: Gregory E. FasshauerAbstract:We propose a fast and accurate Approximation method for large sets of multivariate data using radial functions. In the tradi- tional radial basis function approach this task is usually accomplished by solving a large system of linear equations stemming from an inter- polation formulation. In the traditional moving Least-Squares method one needs to solve a small linear system for each evaluation of the ap- proximant. We present an Approximation scheme { based on the work on approximate Approximation by Maz'ya and Schmidt { that has ap- proximation properties similar to the moving Least-Squares method, but completely avoids the solution of linear systems. Moreover, the sums required for the evaluation of the approximant can be processed quickly. We establish a connection to traditional radial basis func- tion Approximation by using appropriate radial generating functions. Examples of locally supported as well as globally supported functions with arbitrary Approximation orders are given.
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Iterated Approximate Moving Least Squares Approximation
Advances in Meshfree Techniques, 1Co-Authors: Gregory E. Fasshauer, Jack G. ZhangAbstract:The radial basis function interpolant is known to be the best Approximation to a set of scattered data when the error is measured in the native space norm. The approximate moving Least Squares method, on the other hand, was recently proposed as an efficient Approximation method that avoids the solution of the system of linear equations associated with the radial basis function interpolant. In this paper we propose and analyze an algorithm that iterates on the residuals of an approximate moving Least Squares Approximation. We show that this algorithm yields the radial basis interpolant in the limit. Supporting numerical experiments are also included.
Chenlei Leng - One of the best experts on this subject based on the ideXlab platform.
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Least Squares Approximation with a diverging number of parameters
Statistics & Probability Letters, 2009Co-Authors: Chenlei LengAbstract:Abstract Regularized regression with the l 1 penalty is a popular approach for variable selection and coefficient estimation. For a unified treatment of the l 1 -constrained model selection, Wang and Leng (2007) proposed the Least Squares Approximation method (LSA) for a fixed dimension. LSA makes use of a quadratic expansion of the loss function and takes full advantage of the fast Lasso algorithm in Efron et al. (2004) . In this paper, we extend the fixed dimension LSA to the situation with a diverging number of parameters. We show that LSA possesses the oracle properties under appropriate conditions when the number of variables grows with the sample size. We propose a new tuning parameter selection method which achieves the oracle properties. Extensive simulation studies confirmed the theoretical results.
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Unified LASSO Estimation by Least Squares Approximation
Journal of the American Statistical Association, 2007Co-Authors: Hansheng Wang, Chenlei LengAbstract:We propose a method of Least Squares Approximation (LSA) for unified yet simple LASSO estimation. Our general theoretical framework includes ordinary Least Squares, generalized linear models, quantile regression, and many others as special cases. Specifically, LSA can transfer many different types of LASSO objective functions into their asymptotically equivalent Least Squares problems. Thereafter, the standard asymptotic theory can be established and the LARS algorithm can be applied. In particular, if the adaptive LASSO penalty and a Bayes information criterion–type tuning parameter selector are used, the resulting LSA estimator can be as efficient as the oracle. Extensive numerical studies confirm our theory.
Marco Vianello - One of the best experts on this subject based on the ideXlab platform.
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Geometric weakly admissible meshes, discrete Least Squares Approximations and approximate Fekete points
Mathematics of Computation, 2011Co-Authors: Len Bos, Jean-paul Calvi, Norman Levenberg, Alvise Sommariva, Marco VianelloAbstract:Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate Least Squares Approximation and Lagrange interpolation.
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Geometric Weakly Admissible Meshes, Discrete Least Squares Approximations and Approximate Fekete Points
arXiv: Numerical Analysis, 2009Co-Authors: Len Bos, Jean-paul Calvi, Norman Levenberg, Alvise Sommariva, Marco VianelloAbstract:Using the concept of Geometric Weakly Admissible Meshes together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate Least Squares Approximation and Lagrange interpolation.