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UTV decomposition of dual matrices and its applications

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Abstract

Matrix factorization in the context of dual numbers has found applications, in recent years, in fields such as kinematics and computer graphics. In this paper, we develop an efficient approach for handling large-scale data low-rank approximation problems using the UTV decomposition of dual matrices (DUTV). Theoretically, we propose an explicit expression for the DUTV and provide necessary and sufficient conditions for its existence. During this process, we also discovered that the general low-rank model for dual matrices can be solved by the Sylvester equation. In numerical experiments, the DUTV algorithm outperforms the dual matrix SVD algorithm in terms of speed and maintains effective performance in wave recognition. Subsequently, we utilize the DUTV algorithm to validate brain functional circuits in large-scale task-state functional magnetic resonance imaging data. Successfully identifying three types of wave signals, the DUTV method provides substantial empirical evidence for cognitive neuroscience theories.

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Notes

  1. https://www.humanconnectome.org/storage/app/media/documentation/s1200/HCP_S1200_Release_Reference_Manual.pdf.

  2. http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf.

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Acknowledgements

The authors would like to thank the managing editor and two referees for their very detailed comments, which have significantly improved the presentation of our manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Renjie Xu

  2. Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, 200433, People’s Republic of China

    Tong Wei

  3. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

  4. Department of Electrical Engineering and Center for Intelligent Multidimensional Data Analysis, City University of Hong Kong, Kowloon, Hong Kong, People’s Republic of China

    Hong Yan

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  1. Renjie Xu
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Correspondence to Yimin Wei.

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Communicated by Wei Gong.

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R. Xu is supported by the National Natural Science Foundation of China under Grant 12271108 and Shanghai Municipal Science and Technology Commission under Grant 23WZ2501400. T. Wei is partially supported by the Science and Technology Commission of Shanghai Municipality (No. 23ZR1403000, 20JC1419500, 2018SHZDZX0). Y. Wei is supported by the National Natural Science Foundation of China under Grant 12271108, the Ministry of Science and Technology of China under Grant G2023132005L and Medical Engineering Joint Fund of Fudan University. H. Yan is supported by the Hong Kong Research Grants Council (Project 11204821), the Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA) and City University of Hong Kong (Projects 9610034 and 9610460)

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Xu, R., Wei, T., Wei, Y. et al. UTV decomposition of dual matrices and its applications. Comp. Appl. Math. 43, 41 (2024). https://doi.org/10.1007/s40314-023-02565-7

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  • DOI: https://doi.org/10.1007/s40314-023-02565-7

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