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An inexactly accelerated algorithm for nonnegative tensor CP decomposition with the column unit constraints

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Abstract

The component separation problem in complex chemical systems is very important and challenging in chemometrics. In this paper, we study a third-order nonnegative CANDECOMP/PARAFAC decomposition model with the column unit constraints (NCPD_CU) motivated by the component separation problem. To solve the NCPD_CU model, we first explore rapid computational methods for a generalized class of three-block optimization problems, which may exhibit nonconvexity and nonsmoothness. To this end, we propose the accelerated inexact block coordinate descent (AIBCD) algorithm, where each subproblem is inexactly solved through a finite number of inner-iterations employing the alternating proximal gradient method. Additionally, the algorithm incorporates extrapolation during the outer-iterations to enhance overall efficiency. We prove that the iterative sequence generated by the algorithm converges to a stationary point under mild conditions. Subsequently, we apply this methodology to the NCPD_CU model that satisfies the specified conditions. Finally, we present numerical results using both synthetic and real-world data, showcasing the remarkable efficiency of our proposed method.

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Data availibility

The links of the hyperspectral images data and video data that are analyzed in Sect. 5.5 have been included where they are mentioned. For the other data analyzed in Sect. 5.4.2, they are available from the corresponding author on reasonable request.

Notes

  1. https://www.tensorlabplus.net/ .

  2. https://personalpages.manchester.ac.uk/staff/d.h.foster/Local_Illumination_HSIs/Local_Illumination_HSIs_2015.html.

  3. https://www.kaggle.com/datasets/wangzhz/video-data-for-test.

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Acknowledgements

The authors would like to express their gratitude to Professor Hailong Wu and Tong Wang for generously sharing the code of ATLD and the real experimental data used in [16]. Thanks also go to Professor Deqing Wang for providing the code of iAPG in [15], and to Professor Yangyang Xu for providing the code of APG-1 in [20] and BPL for nonnegative matrix factorization in [21]. And thank Professor Lieven De Lathauwer for sending us the code of PANOC in [14].

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  1. School of Mathematics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Zihao Wang & Minru Bai

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  1. Zihao Wang
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Minru Bai: Research supported in part by the National Natural Science Foundation of China under Grant 11971159 and 12071399.

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Wang, Z., Bai, M. An inexactly accelerated algorithm for nonnegative tensor CP decomposition with the column unit constraints. Comput Optim Appl 88, 923–962 (2024). https://doi.org/10.1007/s10589-024-00574-8

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