For Hermitian matrices, the optimal block subspace expansion that minimizes all principal angles to a target invariant subspace is explicitly characterized, with convergence bounds and computable Rayleigh-Ritz versions.
Improved Analyses of the Randomized Power Method and Block Lanczos Method
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abstract
The power method and block Lanczos method are popular numerical algorithms for computing the truncated singular value decomposition (SVD) and eigenvalue decomposition problems. Especially in the literature of randomized numerical linear algebra, the power method is widely applied to improve the quality of randomized sketching, and relative-error bounds have been well established. Recently, Musco & Musco (2015) proposed a block Krylov subspace method that fully exploits the intermediate results of the power iteration to accelerate convergence. They showed spectral gap-independent bounds which are stronger than the power method by order-of-magnitude. This paper offers novel error analysis techniques and significantly improves the bounds of both the randomized power method and the block Lanczos method. This paper also establishes the first gap-independent bound for the warm-start block Lanczos method.
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Block subspace expansions for eigenvalues and eigenvectors approximation
For Hermitian matrices, the optimal block subspace expansion that minimizes all principal angles to a target invariant subspace is explicitly characterized, with convergence bounds and computable Rayleigh-Ritz versions.