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<p>Generalized singular values (GSVs) play an essential role in comparative analysis, scientific computing, etc. In the real world data for comparative analysis, both data matrices are usually numerically low-rank. This paper proposes a randomized algorithm that first approximately extracts bases to project the original matrix into a lower-dimensional subspace. It then efficiently calculates the generalized singular value decomposition (GSVD). The accuracy of both basis extraction and comparative analysis quantities, angular distances, generalized fractions of the eigenexpression, and generalized normalized Shannon entropy, are rigorously 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.</p>

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Fast Randomized Algorithms for Low-Rank Matrix Approximations for Generalized Singular Value Decomposition with Applications

  • 期刊论文

  • 发表日期：

    2026年3月27日

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  • Weiwei Xu,
  • Weijie Shen,
  • Wen Li,
  • Weiguo Gao,
  • Yingzhou Li

摘要

Generalized singular values (GSVs) play an essential role in comparative analysis, scientific computing, etc. In the real world data for comparative analysis, both data matrices are usually numerically low-rank. This paper proposes a randomized algorithm that first approximately extracts bases to project the original matrix into a lower-dimensional subspace. It then efficiently calculates the generalized singular value decomposition (GSVD). The accuracy of both basis extraction and comparative analysis quantities, angular distances, generalized fractions of the eigenexpression, and generalized normalized Shannon entropy, are rigorously 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.