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Low-rank matrix estimation via nonconvex optimization methods in multi-response errors-in-variables regression

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  • Xin Li

    (Northwest University)

  • Dongya Wu

    (Northwest University)

Abstract

Noisy and missing data cannot be avoided in real application, such as bioinformatics, economics and remote sensing. Existing methods mainly focus on linear errors-in-variables regression, while relatively little attention is paid for the case of multivariate responses, and how to achieve consistent estimation under corrupted covariates is still an open question. In this paper, a nonconvex error-corrected estimator is proposed for the matrix estimation problem in the multi-response errors-in-variables regression model. Statistical and computational properties for global solutions of the estimator are analysed. In the statistical aspect, the nonasymptotic recovery bound for all global solutions of the nonconvex estimator is established. In the computational aspect, the proximal gradient method is applied to solve the nonconvex optimization problem and proved to linearly converge to a near-global solution. Sufficient conditions are verified in order to obtain probabilistic consequences for specific types of measurement errors by virtue of random matrix analysis. Finally, simulation results on synthetic and real neuroimaging data demonstrate the theoretical properties and show nice consistency under high-dimensional scaling.

Suggested Citation

  • Xin Li & Dongya Wu, 2024. "Low-rank matrix estimation via nonconvex optimization methods in multi-response errors-in-variables regression," Journal of Global Optimization, Springer, vol. 88(1), pages 79-114, January.
    • Handle: RePEc:spr:jglopt:v:88:y:2024:i:1:d:10.1007_s10898-023-01293-w
    DOI: 10.1007/s10898-023-01293-w
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    References listed on IDEAS

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    1. Alexandre Belloni & Mathieu Rosenbaum & Alexandre B. Tsybakov, 2017. "Linear and conic programming estimators in high dimensional errors-in-variables models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 939-956, June.
    2. Wu, Jie & Zheng, Zemin & Li, Yang & Zhang, Yi, 2020. "Scalable interpretable learning for multi-response error-in-variables regression," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    3. McGillivray, Annaliza & Khalili, Abbas & Stephens, David A., 2020. "Estimating sparse networks with hubs," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    4. Li, Mengyan & Li, Runze & Ma, Yanyuan, 2021. "Inference in high dimensional linear measurement error models," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    5. Hua Zhou & Lexin Li & Hongtu Zhu, 2013. "Tensor Regression with Applications in Neuroimaging Data Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 540-552, June.
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