A Block Bidiagonalization Method for Fixed-Precision Low-Rank Matrix Approximation
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We present randUBV, a randomized algorithm for matrix sketching based on the block Lanzcos bidiagonalization process. Given a matrix A, it produces a low-rank approximation of the form UBV^T, where U and V are approximately orthogonal and B is block bidiagonal. It is closely related to the randQB algorithms of Yu, Gu, and Li (2018) in that the entries of B are incrementally generated and the Frobenius norm approximation error may be efficiently estimated. Our algorithm is therefore suitable for the fixed-precision problem, and so is designed to terminate as soon as a user input error tolerance is reached. Numerical experiments suggest that the block Lanczos method can be competitive with or superior to algorithms that use power iteration, even when A has significant clusters of singular values.
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