A Block Bidiagonalization Method for Fixed-Precision Low-Rank Matrix Approximation

We present randUBV, a randomized algorithm for matrix sketching based on the block Lanzcos bidiagonalization process. Given a matrix $\bf{A}$, it produces a low-rank approximation of the form ${\bf UBV}^T$, where $\bf{U}$ and $\bf{V}$ are approximately orthogonal and $\bf{B}$ is block bidiagonal. It is closely related to the randQB algorithms of Yu, Gu, and Li (2018) in that the entries of $\bf{B}$ 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 $\bf{A}$ has significant clusters of singular values.