  Generalized singular values (GSVs) play an essential role in the comparative
analysis. In the real world data for comparative analysis, both data matrices
are usually numerically low-rank. This paper proposes a randomized algorithm to
first approximately extract bases and then calculate GSVs efficiently. The
accuracy of both basis extration and comparative analysis quantities, angular
distances, generalized fractions of the eigenexpression, and generalized
normalized Shannon entropy, are rigursly analyzed. The proposed algorithm is
applied to both synthetic data sets and the genome-scale expression data sets.
Comparing to other GSVs algorithms, the proposed algorithm achieves the fastest
runtime while preserving sufficient accuracy in comparative analysis.
