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The power method and block Lanczos method are popular numerical algorithmsfor computing the truncated singular value decomposition (SVD) and eigenvaluedecomposition problems. Especially in the literature of randomized numericallinear algebra, the power method is widely applied to improve the quality ofrandomized sketching, and relative-error bounds have been well established.Recently, Musco & Musco (2015) proposed a block Krylov subspace method thatfully exploits the intermediate results of the power iteration to accelerateconvergence. They showed spectral gap-independent bounds which are strongerthan the power method by order-of-magnitude. This paper offers novel erroranalysis techniques and significantly improves the bounds of both therandomized power method and the block Lanczos method. This paper alsoestablishes the first gap-independent bound for the warm-start block Lanczosmethod.

Improved Analyses of the Randomized Power Method and Block Lanczos Method