20,150 research outputs found
Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem
We propose a new algorithm for sparse estimation of eigenvectors in
generalized eigenvalue problems (GEP). The GEP arises in a number of modern
data-analytic situations and statistical methods, including principal component
analysis (PCA), multiclass linear discriminant analysis (LDA), canonical
correlation analysis (CCA), sufficient dimension reduction (SDR) and invariant
co-ordinate selection. We propose to modify the standard generalized orthogonal
iteration with a sparsity-inducing penalty for the eigenvectors. To achieve
this goal, we generalize the equation-solving step of orthogonal iteration to a
penalized convex optimization problem. The resulting algorithm, called
penalized orthogonal iteration, provides accurate estimation of the true
eigenspace, when it is sparse. Also proposed is a computationally more
efficient alternative, which works well for PCA and LDA problems. Numerical
studies reveal that the proposed algorithms are competitive, and that our
tuning procedure works well. We demonstrate applications of the proposed
algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA and SDR.
Supplementary materials are available online
The Anderson model of localization: a challenge for modern eigenvalue methods
We present a comparative study of the application of modern eigenvalue
algorithms to an eigenvalue problem arising in quantum physics, namely, the
computation of a few interior eigenvalues and their associated eigenvectors for
the large, sparse, real, symmetric, and indefinite matrices of the Anderson
model of localization. We compare the Lanczos algorithm in the 1987
implementation of Cullum and Willoughby with the implicitly restarted Arnoldi
method coupled with polynomial and several shift-and-invert convergence
accelerators as well as with a sparse hybrid tridiagonalization method. We
demonstrate that for our problem the Lanczos implementation is faster and more
memory efficient than the other approaches. This seemingly innocuous problem
presents a major challenge for all modern eigenvalue algorithms.Comment: 16 LaTeX pages with 3 figures include
Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum
The computation of eigenvalues of large-scale matrices arising from finite
element discretizations has gained significant interest in the last decade.
Here we present a new algorithm based on slicing the spectrum that takes
advantage of the rank structure of resolvent matrices in order to compute m
eigenvalues of the generalized symmetric eigenvalue problem in operations, where is a small constant
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