32,808 research outputs found
Low-rank approximate inverse for preconditioning tensor-structured linear systems
In this paper, we propose an algorithm for the construction of low-rank
approximations of the inverse of an operator given in low-rank tensor format.
The construction relies on an updated greedy algorithm for the minimization of
a suitable distance to the inverse operator. It provides a sequence of
approximations that are defined as the projections of the inverse operator in
an increasing sequence of linear subspaces of operators. These subspaces are
obtained by the tensorization of bases of operators that are constructed from
successive rank-one corrections. In order to handle high-order tensors,
approximate projections are computed in low-rank Hierarchical Tucker subsets of
the successive subspaces of operators. Some desired properties such as symmetry
or sparsity can be imposed on the approximate inverse operator during the
correction step, where an optimal rank-one correction is searched as the tensor
product of operators with the desired properties. Numerical examples illustrate
the ability of this algorithm to provide efficient preconditioners for linear
systems in tensor format that improve the convergence of iterative solvers and
also the quality of the resulting low-rank approximations of the solution
A Direct Estimation Approach to Sparse Linear Discriminant Analysis
This paper considers sparse linear discriminant analysis of high-dimensional
data. In contrast to the existing methods which are based on separate
estimation of the precision matrix \O and the difference \de of the mean
vectors, we introduce a simple and effective classifier by estimating the
product \O\de directly through constrained minimization. The
estimator can be implemented efficiently using linear programming and the
resulting classifier is called the linear programming discriminant (LPD) rule.
The LPD rule is shown to have desirable theoretical and numerical properties.
It exploits the approximate sparsity of \O\de and as a consequence allows
cases where it can still perform well even when \O and/or \de cannot be
estimated consistently. Asymptotic properties of the LPD rule are investigated
and consistency and rate of convergence results are given. The LPD classifier
has superior finite sample performance and significant computational advantages
over the existing methods that require separate estimation of \O and \de.
The LPD rule is also applied to analyze real datasets from lung cancer and
leukemia studies. The classifier performs favorably in comparison to existing
methods.Comment: 39 pages.To appear in Journal of the American Statistical Associatio
H-matrix accelerated second moment analysis for potentials with rough correlation
We consider the efficient solution of partial differential equationsfor strongly elliptic operators with constant coefficients and stochastic Dirichlet data by the boundary integral equation method. The computation of the solution's two-point correlation is well understood if the two-point correlation of the Dirichlet data is known and sufficiently smooth.Unfortunately, the problem becomes much more involved in case of rough data. We will show that the concept of the H-matrix arithmetic provides a powerful tool to cope with this problem. By employing a parametric surface representation, we end up with an H-matrix arithmetic based on balanced cluster trees. This considerably simplifies the implementation and improves the performance of the H-matrix arithmetic. Numerical experiments are provided to validate and quantify the presented methods and algorithms
Statistical Inferences Using Large Estimated Covariances for Panel Data and Factor Models
While most of the convergence results in the literature on high dimensional
covariance matrix are concerned about the accuracy of estimating the covariance
matrix (and precision matrix), relatively less is known about the effect of
estimating large covariances on statistical inferences. We study two important
models: factor analysis and panel data model with interactive effects, and
focus on the statistical inference and estimation efficiency of structural
parameters based on large covariance estimators. For efficient estimation, both
models call for a weighted principle components (WPC), which relies on a high
dimensional weight matrix. This paper derives an efficient and feasible WPC
using the covariance matrix estimator of Fan et al. (2013). However, we
demonstrate that existing results on large covariance estimation based on
absolute convergence are not suitable for statistical inferences of the
structural parameters. What is needed is some weighted consistency and the
associated rate of convergence, which are obtained in this paper. Finally, the
proposed method is applied to the US divorce rate data. We find that the
efficient WPC identifies the significant effects of divorce-law reforms on the
divorce rate, and it provides more accurate estimation and tighter confidence
intervals than existing methods
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