Abstract
Nonnegative tensor ring (NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method (NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation (LRA) is introduced to NTR-MM (named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.
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References
Liu W, Zheng N, You Q. Nonnegative matrix factorization and its applications in pattern recognition. Chin Sci Bull, 2006, 51: 7–18
Lee D D, Seung H S. Learning the parts of objects by non-negative matrix factorization. Nature, 1999, 401: 788–791
Zhou G, Cichocki A, Zhao Q, et al. Nonnegative matrix and tensor factorizations: An algorithmic perspective. IEEE Signal Process Mag, 2014, 31: 54–65
Zhou G, Cichocki A, Zhao Q, et al. Efficient nonnegative tucker decompositions: Algorithms and uniqueness. IEEE Trans Image Process, 2015, 24: 4990–5003
Zhou G X, Yang Z Y, Xie S L, et al. Online blind source separation using incremental nonnegative matrix factorization with volume constraint. IEEE Trans Neural Netw, 2011, 22: 550–560
Zhou G, Zhao Q, Zhang Y, et al. Linked component analysis from matrices to high-order tensors: Applications to biomedical data. Proc IEEE, 2016, 104: 310–331
Zhou G, Cichocki A, Zhang Y, et al. Group component analysis for multiblock data: Common and individual feature extraction. IEEE Trans Neural Netw Learning Syst, 2016, 27: 2426–2439
Liang N, Yang Z, Li Z, et al. Semi-supervised multi-view clustering with graph-regularized partially shared non-negative matrix factorization. Knowledge-Based Syst, 2020, 190: 105185
Liang N, Yang Z, Li Z, et al. Multi-view clustering by non-negative matrix factorization with co-orthogonal constraints. Knowledge-Based Syst, 2020, 194: 105582
Jia Y, Kwong S, Hou J, et al. Semi-supervised non-negative matrix factorization with dissimilarity and similarity regularization. IEEE Trans Neural Netw Learning Syst, 2020, 31: 2510–2521
Shang F, Jiao L C, Wang F. Graph dual regularization non-negative matrix factorization for co-clustering. Pattern Recognition, 2012, 45: 2237–2250
He W, Yao Q, Li C, et al. Non-local meets global: An integrated paradigm for hyperspectral image restoration. IEEE Trans Pattern Anal Mach Intell, 2020, doi: https://doi.org/10.1109/TPAMI.2020.3027563
Li J, Zhou G, Qiu Y, et al. Deep graph regularized non-negative matrix factorization for multi-view clustering. Neurocomputing, 2020, 390: 108–116
Chen X, Zhou G, Wang Y, et al. Accommodating multiple tasks’ disparities with distributed knowledge-sharing mechanism. IEEE Trans Cybern, 2020, doi: https://doi.org/10.1109/TCYB.2020.3002911
Yu J, Zhou G, Li C, et al. Low tensor-ring rank completion by parallel matrix factorization. IEEE Trans Neural Netw Learning Syst, 2021, 32: 3020–3033
Wang K, Gu X F, Yu T, et al. Classification of hyperspectral remote sensing images using frequency spectrum similarity. Sci China Tech Sci, 2013, 56: 980–988
Hu W, Tao D, Zhang W, et al. The twist tensor nuclear norm for video completion. IEEE Trans Neural Netw Learning Syst, 2017, 28: 2961–2973
Cui G C, Zhu L, Gui L H, et al. Multidimensional clinical data de-noising via Bayesian CP factorization. Sci China Tech Sci, 2020, 63: 249–254
Huang Z, Qiu Y, Sun W. Recognition of motor imagery EEG patterns based on common feature analysis. Brain-Comput Interfaces, 2020, 1: 1–9
Qiu Y, Zhou G, Wang Y, et al. A generalized graph regularized nonnegative tucker decomposition framework for tensor data representation. IEEE Trans Cybern, 2020, doi: https://doi.org/10.1109/TCYB.2020.2979344
Qiu Y, Zhou G, Zhang Y, et al. Graph regularized nonnegative tucker decomposition for tensor data representation. In: Proceedings of the ICASSP 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Brighton: IEEE, 2019. 8613–8617
Yu Y, Zhou G, Zheng N, et al. Graph regularized nonnegative tensor ring decomposition for multiway representation learning. arXiv: 2010.05657
Zhao Q, Zhou G, Xie S, et al. Tensor ring decomposition. arXiv: 1606.05535
Zhao Q, Sugiyama M, Yuan L, et al. Learning efficient tensor representations with ring-structured networks. In: Proceedings of the ICASSP 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Brighton: IEEE, 2019. 8608–8612
Lee D, Seung H S. Algorithms for non-negative matrix factorization. Adv Neural Info Process Syst, 2000, 13: 556–562
Cichocki A, Zdunek R, Amari S I. Hierarchical als algorithms for non-negative matrix and 3d tensor factorization. In: Proceedings of the International Conference on Independent Component Analysis and Signal Separation. Springer, 2007. 169–176
Guan N, Tao D, Luo Z, et al. NeNMF: An optimal gradient method for nonnegative matrix factorization. IEEE Trans Signal Process, 2012, 60: 2882–2898
Zheng N, Hayami K, Yin J F. Modulus-type inner outer iteration methods for nonnegative constrained least squares problems. SIAM J Matrix Anal Appl, 2016, 37: 1250–1278
Dong J L, Jiang M Q. A modified modulus method for symmetric positive-definite linear complementarity problems. Numer Linear Algebra Appl, 2009, 16: 129–143
Bai Z Z. Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer Linear Algebra Appl, 2010, 17: 917–933
Zhang Y, Zhou G, Zhao Q, et al. Fast nonnegative tensor factorization based on accelerated proximal gradient and low-rank approximation. Neurocomputing, 2016, 198: 148–154
Zhou G, Cichocki A, Xie S. Fast nonnegative matrix/tensor factorization based on low-rank approximation. IEEE Trans Signal Process, 2012, 60: 2928–2940
Wang P, He Z, Xie K, et al. A hybrid algorithm for low-rank approximation of nonnegative matrix factorization. Neurocomputing, 2019, 364: 129–137
Paatero P. The multilinear enginea table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model. J Comput Graph Stat, 1999, 8: 854–888
Kolda T G, Bader B W. Tensor decompositions and applications. SIAM Rev, 2009, 51:455–500
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 62073087, 61973087 and 61973090), and the Key-Area Research and Development Program of Guangdong Province (Grant No. 2019B010154002).
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Yu, Y., Xie, K., Yu, J. et al. Fast nonnegative tensor ring decomposition based on the modulus method and low-rank approximation. Sci. China Technol. Sci. 64, 1843–1853 (2021). https://doi.org/10.1007/s11431-020-1820-x
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DOI: https://doi.org/10.1007/s11431-020-1820-x