  In this paper we tackle two important challenges related to the accurate
partial singular value decomposition (SVD) and numerical rank estimation of a
huge matrix to use in low-rank learning problems in a fast way. We use the
concepts of Krylov subspaces such as the Golub-Kahan bidiagonalization process
as well as Ritz vectors to achieve these goals. Our experiments identify
various advantages of the proposed methods compared to traditional and
randomized SVD (R-SVD) methods with respect to the accuracy of the singular
values and corresponding singular vectors computed in a similar execution time.
The proposed methods are appropriate for applications involving huge matrices
where accuracy in all spectrum of the desired singular values, and also all of
corresponding singular vectors is essential. We evaluate our method in the real
application of Riemannian similarity learning (RSL) between two various image
datasets of MNIST and USPS.
