Randomized algorithms for low-rank matrix approximation are investigated,
with the emphasis on the fixed-precision problem and computational efficiency
for handling large matrices. The algorithms are based on the so-called QB
factorization, where Q is an orthonormal matrix. Firstly, a mechanism for
calculating the approximation error in Frobenius norm is proposed, which
enables efficient adaptive rank determination for large and/or sparse matrix.
It can be combined with any QB-form factorization algorithm in which B's rows
are incrementally generated. Based on the blocked randQB algorithm by P.-G.
Martinsson and S. Voronin, this results in an algorithm called randQB EI. Then,
we further revise the algorithm to obtain a pass-efficient algorithm, randQB
FP, which is mathematically equivalent to the existing randQB algorithms and
also suitable for the fixed-precision problem. Especially, randQB FP can serve
as a single-pass algorithm for calculating leading singular values, under
certain condition. With large and/or sparse test matrices, we have empirically
validated the merits of the proposed techniques, which exhibit remarkable
speedup and memory saving over the blocked randQB algorithm. We have also
demonstrated that the single-pass algorithm derived by randQB FP is much more
accurate than an existing single-pass algorithm. And with data from a scenic
image and an information retrieval application, we have shown the advantages of
the proposed algorithms over the adaptive range finder algorithm for solving
the fixed-precision problem.Comment: 21 pages, 10 figure
