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Regularized matrix regression

Author

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  • Hua Zhou
  • Lexin Li

Abstract

type="main" xml:id="rssb12031-abs-0001"> Modern technologies are producing a wealth of data with complex structures. For instance, in two-dimensional digital imaging, flow cytometry and electroencephalography, matrix-type covariates frequently arise when measurements are obtained for each combination of two underlying variables. To address scientific questions arising from those data, new regression methods that take matrices as covariates are needed, and sparsity or other forms of regularization are crucial owing to the ultrahigh dimensionality and complex structure of the matrix data. The popular lasso and related regularization methods hinge on the sparsity of the true signal in terms of the number of its non-zero coefficients. However, for the matrix data, the true signal is often of, or can be well approximated by, a low rank structure. As such, the sparsity is frequently in the form of low rank of the matrix parameters, which may seriously violate the assumption of the classical lasso. We propose a class of regularized matrix regression methods based on spectral regularization. A highly efficient and scalable estimation algorithm is developed, and a degrees-of-freedom formula is derived to facilitate model selection along the regularization path. Superior performance of the method proposed is demonstrated on both synthetic and real examples.

Suggested Citation

  • Hua Zhou & Lexin Li, 2014. "Regularized matrix regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(2), pages 463-483, March.
  • Handle: RePEc:bla:jorssb:v:76:y:2014:i:2:p:463-483
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    File URL: http://hdl.handle.net/10.1111/rssb.2014.76.issue-2
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    Citations

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    Cited by:

    1. Pan Shang & Lingchen Kong, 2021. "Regularization Parameter Selection for the Low Rank Matrix Recovery," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 772-792, June.
    2. Yang, Yaohong & Zhao, Weihua & Wang, Lei, 2023. "Online regularized matrix regression with streaming data," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    3. Xiaoshan Li & Da Xu & Hua Zhou & Lexin Li, 2018. "Tucker Tensor Regression and Neuroimaging Analysis," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 10(3), pages 520-545, December.
    4. Yang, Yaohong & Wang, Lei & Liu, Jiamin & Li, Rui & Lian, Heng, 2023. "Communication-efficient estimation of quantile matrix regression for massive datasets," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    5. Zhao, Junlong & Niu, Lu & Zhan, Shushi, 2017. "Trace regression model with simultaneously low rank and row(column) sparse parameter," Computational Statistics & Data Analysis, Elsevier, vol. 116(C), pages 1-18.
    6. Xu Gao & Weining Shen & Liwen Zhang & Jianhua Hu & Norbert J. Fortin & Ron D. Frostig & Hernando Ombao, 2021. "Regularized matrix data clustering and its application to image analysis," Biometrics, The International Biometric Society, vol. 77(3), pages 890-902, September.
    7. Wang, Lei & Zhang, Jing & Li, Bo & Liu, Xiaohui, 2022. "Quantile trace regression via nuclear norm regularization," Statistics & Probability Letters, Elsevier, vol. 182(C).
    8. Wang, Dong & Liu, Xialu & Chen, Rong, 2019. "Factor models for matrix-valued high-dimensional time series," Journal of Econometrics, Elsevier, vol. 208(1), pages 231-248.
    9. Fijnanda van Klingeren & Vincent Buskens, 2024. "Graduated sanctioning, endogenous institutions and sustainable cooperation in common-pool resources: An experimental test," Rationality and Society, , vol. 36(2), pages 183-229, May.
    10. Wei Hu & Tianyu Pan & Dehan Kong & Weining Shen, 2021. "Nonparametric matrix response regression with application to brain imaging data analysis," Biometrics, The International Biometric Society, vol. 77(4), pages 1227-1240, December.
    11. Amado Villarreal González & Saidi Magaly Flores Sánchez & Miguel A. Flores Segovia, 2016. "Patrones de co-localización espacial de la industria aeroespacial en México," Estudios Económicos, El Colegio de México, Centro de Estudios Económicos, vol. 31(1), pages 169-211.
    12. Yin Xia & Lexin Li, 2017. "Hypothesis testing of matrix graph model with application to brain connectivity analysis," Biometrics, The International Biometric Society, vol. 73(3), pages 780-791, September.
    13. Federico Ferraccioli & Giovanna Menardi, 2023. "Modal clustering of matrix-variate data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(2), pages 323-345, June.
    14. Luo, Chongliang & Liang, Jian & Li, Gen & Wang, Fei & Zhang, Changshui & Dey, Dipak K. & Chen, Kun, 2018. "Leveraging mixed and incomplete outcomes via reduced-rank modeling," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 378-394.
    15. Zengchao Xu & Shan Luo & Zehua Chen, 2023. "A Portmanteau Local Feature Discrimination Approach to the Classification with High-dimensional Matrix-variate Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 441-467, February.
    16. Canu, Stéphane & Fourdrinier, Dominique, 2017. "Unbiased risk estimates for matrix estimation in the elliptical case," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 60-72.
    17. Niu, Lu & Liu, Xiumin & Zhao, Junlong, 2020. "Robust estimator of the correlation matrix with sparse Kronecker structure for a high-dimensional matrix-variate," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    18. Li, Mei & Kong, Lingchen, 2019. "Double fused Lasso penalized LAD for matrix regression," Applied Mathematics and Computation, Elsevier, vol. 357(C), pages 119-138.
    19. Hung Hung & Su‐Yun Huang, 2019. "Sufficient dimension reduction via random‐partitions for the large‐p‐small‐n problem," Biometrics, The International Biometric Society, vol. 75(1), pages 245-255, March.
    20. Philip T. Reiss & Jeff Goldsmith & Han Lin Shang & R. Todd Ogden, 2017. "Methods for Scalar-on-Function Regression," International Statistical Review, International Statistical Institute, vol. 85(2), pages 228-249, August.
    21. Ruonan Li & Luo Xiao, 2023. "Latent factor model for multivariate functional data," Biometrics, The International Biometric Society, vol. 79(4), pages 3307-3318, December.

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