A Krylov Subspace Method for Covariance Approximation and Simulation of Random Processes and Fields
Multidimensional Systems and Signal Processing
CGLS-GCV: a hybrid algorithm for low-rank-deficient problems
Applied Numerical Mathematics - Special issue: 2nd international workshop on numerical linear algebra, numerical methods for partial differential equations and optimization
Generalized consistent estimation on low-rank Krylov subspaces of arbitrarily high dimension
IEEE Transactions on Signal Processing
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This paper considers the problem of finding the principal eigenspace and/or eigenpairs of $M \times M$ Hermitian matrices that can be expressed or approximated by a low-rank matrix plus a shift, i.e., $\bf{A} = \bf{B} + \sigma \bf{I}$, where $\bf{B}$ is a rank $d$ Hermitian matrix and $d \ll M$. Such matrices arise in signal processing, geophysics, dynamic structure analysis, and other fields. The proposed problem can be solved by a full $O(M^3)$ eigendecomposition, or by several more efficient alternatives, e.g., the power, subspace iteration, and Lanczos algorithms. This paper shows that the Lanczos algorithm can exploit the inherent structure and is generally more efficient than other alternatives. More specifically, if $\bf{A} = \bf{B} +\sigma \bf{I}$, the Lanczos algorithm can be used to exactly determine the principal eigenspace span$\bf{B}$ and $\sigma$ with a finite amount of computation. If $\bf{A}$ is close to $\bf{B} + \sigma \bf{I}$, the Lanczos algorithm can estimate the principal eigenvectors and eigenvalues in $O(M^2d)$ flops. It is shown that the errors in the estimates of the $k$th principal eigenvalue $\lambda_k$ and eigenvector $\bf{e}_k$ decay at the rate of $\varepsilon^2 / (\lambda_k - \sigma )^2$ and $\varepsilon /(\lambda _k- \sigma)$, respectively, where $\varepsilon$ is a measure of the mismatch between $\bf{A}$ and $\bf{B} + \sigma \bf{I}$.