Abstract
Semi-supervised multi-view clustering in the subspace has attracted sustained attention. The existing methods often project the samples with the same label into the same point in the low dimensional space. This hard constraint-based method magnifies the dimension reduction error, restricting the subsequent clustering performance. To relax the labeled data during projection, we propose a novel method called label relaxation-based semi-supervised non-negative matrix factorization (LRSNMF). In our method, we first employ the Spearman correlation coefficient to measure the similarity between samples. Based on this, we design a new relaxed non-negative label matrix for better subspace learning, instead of the binary matrix. Also, we derive an updated algorithm based on an alternative iteration rule to solve the proposed model. Finally, the experimental results on three real-world datasets (i.e., MSRC, ORL1, and ORL2) with six evaluation indexes (i.e., accuracy, NMI, purity, F-score, precision, and recall) show the advantages of our LRSNMF, with comparison to the state-of-the-art methods.
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References
Huang, L., Chao, H., Wang, C.: Multi-view intact space clustering. Pattern Recogn. 86, 344–353 (2019). https://doi.org/10.1016/j.patcog.2018.09.016
Zeng, S., Wang, X., Cui, H., Zheng, C., Feng, D.D.: A unified collaborative multi kernel fuzzy clustering for multi-view data. IEEE Trans Fuzzy Syst 26(3), 1671–1687 (2018). https://doi.org/10.1109/TFUZZ.2017.2743679
Wang, S., Zou, Y., Min, W., Xiong, X.: Multi-view face generation via unpaired images. Vis. Comput. (2021). https://doi.org/10.1007/s00371-021-02129-y
Zhan K, Niu C, Chen C, Nie F, Zhang C, Yang Y: Graph structure fusion for multi-view clustering. IEEE Trans Knowl Data Eng, 31(10):1984–1993 (2019),URL https://doi.org/10.1109/TKDE.2018.2872061
Zhan K, Zhang C, Guan J, Wang J: Graph learning for multi-view clustering. IEEE Trans Cybern 48(10):2887–2895 (2018),URL https://doi.org/10.1109/TCYB.2017.2751646
Bhattacharjee SD, Yuan J, Huang Y, Meng J, Duan L: Query adaptive multi-view object instance search and localization using sketches. IEEE Trans Multim 20(10):2761–2773 (2018),URL https://doi.org/10.1109/TMM.2018.2814338
Zhang, Z., Liu, L., Shen, F., Shen, H.T., Shao, L.: Binary multi-view clustering. IEEE Trans Pattern Anal Mach Intell 41(7), 1774–1782 (2019). https://doi.org/10.1109/TPAMI.2018.2847335
Nie, F., Cai, G., Li, J., Li, X.: Auto-weighted multi-view learning for image clustering and semi-supervised classification. IEEE Trans Image Process 27(3), 1501–1511 (2018). https://doi.org/10.1109/TIP.2017.2754939
DaiW, ErdtM, Sourin,A: Self-supervised pairing image clustering for automated quality control. The Visual Computer, 1–14 (2021), URL https://doi.org/10.1007/s00371-021-02137-y
YangZ, ZhangY, XiangY, YanW, XieS: Non-negative matrix factorization with dual constraints for image clustering. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50(7), 2524–2533 (2018), URL https://ieeexplore.ieee.org/document/8360970
Li, J., Yang, B., Yang, W., Sun, C., Xu, J.: Subspace-based multi-view fusion for instance-level image retrieval. Vis. Comput. 37(3), 619–633 (2021). https://doi.org/10.1007/s00371-020-01828-2
ZhanK, NieF.: WangJ, YangY: Multi-view consensus graph clustering. IEEE Trans. Image Process. 28(3), 1261–1270 (2018). https://doi.org/10.1109/TIP.2018.2877335
WangH, YangY.: LiuB: GMC: Graph-based multi-view clustering. IEEE Trans. Knowl. Data Eng. 32(6), 1116–1129 (2019). https://doi.org/10.1109/TKDE.2019.2903810
WangR, NieF.: WangZ, HuH, LiX: Parameter-free weighted multi-view projected clustering with structured graph learning. IEEE Trans. Knowl. Data Eng. 32(10), 2014–2025 (2019). https://doi.org/10.1109/TKDE.2019.2913377
Yang, Z., Zhang, Y., Xiang, Y., Yan, W., Xie, S.: Non-negative matrix factorization with dual constraints for image clustering. IEEE Transactions on Systems, Man, and Cybernetics: Systems 50(7), 2524–2533 (2020). https://doi.org/10.1109/TSMC.2018.2820084
Zong L, Zhang X, Zhao L, Yu H, Zhao Q: Multi-view clustering via multi-manifold regularized non-negative matrix factorization. Neural Networks 88:74–89 (2017),URL https://doi.org/10.1016/j.neunet.2017.02.003
Ou W, Yu S, Li G, Lu J, Zhang K, Xie G: Multi-view non-negative matrix factorization by patch alignment framework with view consistency. Neurocomputing 204:116–124 (2016),URL https://doi.org/10.1016/j.neucom.2015.09.133
Yang Z, Liang N, Yan W, Li Z, Xie S: Uniform distribution non-negative matrix factorization for multi-view clustering. IEEE transactions on cybernetics51(6):3249–3262 (2020),URL https://doi.org/10.1109/TCYB.2020.2984552
Gao J, Han J, Liu J, Wang C: Multi-view clustering via joint non-negative matrix factorization. In: Proceedings of the 13th SIAM International Conference on Data Mining, Austin, Texas, USA, SIAM, pp 252–260 (2013), URL https://doi.org/10.1137/1.9781611972832.28
Huang S, Kang Z, Xu Z: Self-weighted multi-view clustering with soft capped norm. Knowl Based Syst 158:1–8 (2018),URL https://doi.org/10.1016/j.knosys.2018.05.017
Wang H, Yang Y, Li T: Multi-view clustering via concept factorization with local manifold regularization. In: Bonchi F, Domingo-Ferrer J, Baeza-Yates R, Zhou Z, Wu X (eds) IEEE 16th International Conference on Data Mining, Barcelona, Spain, IEEE Computer Society, pp 1245–1250 (2016),URL https://doi.org/10.1109/ICDM.2016.0167
Shao W, He L, Lu C, Wei X, Yu PS: Online unsupervised multi-view feature selection. In: Bonchi F, Domingo-Ferrer J, Baeza-Yates R, Zhou Z, Wu X (eds) IEEE 16th International Conference on Data Mining, Barcelona, Spain, IEEE Computer Society, pp 1203–1208 (2016), URL https://doi.org/10.1109/ICDM.2016.0160
Tan Y, Ou W, Long F, Wang P, Xue Y: Multi-view clustering via co-regularized nonnegative matrix factorization with correlation constraint. In: 7th International Conference on Cloud Computing and Big Data, Macau, China, IEEE Computer Society, pp 1–6 (2016),URL https://doi.org/10.1109/CCBD.2016.012
Wang, J., Tian, F., Yu, H., Liu, C.H., Zhan, K., Wang, X.: Diverse non-negative matrix factorization for multi-view data representation. IEEE Trans Cybern 48(9), 2620–2632 (2018). https://doi.org/10.1109/TCYB.2017.2747400
Wang, X., Zhang, T., Gao, X.: Multiview clustering based on non-negative matrix factorization and pairwise measurements. IEEE Trans Cybern 49(9), 3333–3346 (2019). https://doi.org/10.1109/TCYB.2018.2842052
ZhaoH, DingZ, FuY: Multi-view clustering via deep matrix factorization. In Thirty-First AAAI Conference on Artificial Intelligence. 2921–2927 (2017), URL http://aaai.org/ocs/index.php/AAAI/AAAI17/paper/view/14647
YiY, ShiY, ZhangH, WangJ, KongJ: Label propagation based semi-supervised non-negative matrix factorization for feature extraction. Neurocomputing, 1491021–1037 (2015), URL https://doi.org/10.1016/j.neucom.2014.07.031
QiuY, HaoP.: Self-supervised deep subspace clustering network for faces in videos. Vis. Comput. 37(8), 2253–2261 (2021). https://doi.org/10.1007/s00371-020-01984-5
Qian, B., Gu, X., Shu, Z., Shen X: Constrained multi-view nmf with graph embedding for face clustering.: 17th International Symposium on Distributed Computing and Applications for Business Engineering and Science (DCABES), pp 103–106 (2018). URL (2018). https://doi.org/10.1109/DCABES.2018.00036
Li, G., Zhang, X., Zheng, S., Li, D.: Semi-supervised convex nonnegative matrix factorizations with graph regularized for image representation. Neurocomputing 237, 1–11 (2017). https://doi.org/10.1016/j.neucom.2016.04.028
Cai, H., Liu, B., Xiao, Y., Lin, L.: Semi-supervised multi-view clustering based on orthonormalityconstrained nonnegative matrix factorization. Inf S- ci 536, 171–184 (2020). https://doi.org/10.1016/j.ins.2020.05.073
Liu, J., Jiang, Y., Li, Z., Zhou, Z., Lu, H.: Partially shared latent factor learning with multi-view data. IEEE Trans Neural Networks Learn Syst 26(6), 1233–1246 (2015). https://doi.org/10.1109/TNNLS.2014.2335234
Jiang, Y., Liu, J., Li, Z., Lu, H.: Semi-supervised unified latent factor learning with multi-view data. Mach Vis Appl 25(7), 1635–1645 (2014). https://doi.org/10.1007/s00138-013-0556-3
Peng, S., Ser, W., Chen, B., Lin, Z.: Robust semi-supervised nonnegative matrix factorization for image clustering. Pattern Recogn. 111, 107683 (2021). https://doi.org/10.1016/j.patcog.2020.107683
Xing, Z., Wen, M., Peng, J., Feng, J.: Discriminative semi-supervised non-negative matrix factorization for data clustering. Eng Appl Artif Intell 103, 104289 (2021). https://doi.org/10.1016/j.engappai.2021.104289
Wang J, Wang X, Tian F, Liu CH, Yu H, Liu Y: Adaptive multi-view semi-supervised nonnegative matrix factorization. In: Hirose A, Ozawa S, Doya K, Ikeda K, Lee M, Liu D (eds) Neural Information Processing -23rd International Conference, Kyoto, Japan, Lecture Notes in Computer Science, vol 9948, pp 435–444 (2016), URL https://doi.org/10.1007/978-3-319-46672-9_49
Shi, S., Nie, F., Wang, R., Li, X.: Auto-weighted multi-view clustering via spectral embedding. Neurocomputing 399, 369–379 (2020). https://doi.org/10.1016/j.neucom.2020.02.071
Nie F, Li J, Li X: Self-weighted Multiview Clustering with Multiple Graphs. Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence IJCAI:2564–2570 (2017), URL https://doi.org/10.24963/ijcai.2017/357
Nie F, Li J, Li X: Parameter-Free Auto-Weighted Multiple Graph Learning: A Framework for Multiview Clustering and Semi-Supervised Classification. Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence 1881–1887 (2016), URL http://www.ijcai.org/Abstract/16/269
Nie, F., Shi, S., Li, X.: Auto-weighted multi-view co-clustering via fast matrix factorization. Pattern Recogn. 102, 107207 (2020). https://doi.org/10.1016/j.patcog.2020.107207
Fang, X., Xu, Y., Li, X., Lai, Z., Wong, W., Fang, B.: Regularized label relaxation linear regression. IEEE transactions on neural networks and learning systems 29(4), 1006–1018 (2017). https://doi.org/10.1109/TNNLS.2017.2648880
Xiang, S., Nie, F., Meng, G., Pan, C., Zhang, C.: Discriminative least squares regression for multiclass classification and feature selection. IEEE Trans Neural Networks Learn Syst 23(11), 1738–1754 (2012). https://doi.org/10.1109/TNNLS.2012.2212721
Zhang, W., Wei, Z., Wang, B., Han, X.: Commentary: Measuring mixing patterns in complex networks by Spearman rank correlation coefficient. Physica A 451, 440–450 (2016). https://doi.org/10.1016/j.physa.2016.01.056
Yang, Z., Xiang, Y., Xie, K., Lai, Y.: Adaptive method for non-smooth non- negative matrix factorization. IEEE Transactions on Neural Networks and Learning Systems 28(4), 948–960 (2017). https://doi.org/10.1109/TNNLS.2016.2517096
Yang, Z., Zhou, G., Xie, S., Ding, S., Yang, J., Zhang, J.: Blind spectral un- mixing based on sparse nonnegative matrix factorization. IEEE Trans Image Process 20(4), 1112–1125 (2011). https://doi.org/10.1109/TIP.2010.2081678
Paatero, P., Tapper, U.: Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values. Environmetrics 5(2), 111–126 (1994). https://doi.org/10.1002/env.3170050203
Cai, D., He, X., Han, J., Huang, T.S.: Graph regularized nonnegative matrix factorization for data representation. IEEE Trans Pattern Anal Mach Intell 33(8), 1548–1560 (2011). https://doi.org/10.1109/TPAMI.2010.231
Ding CHQ, Li T, Peng W, Park H: Orthogonal nonnegative matrix t-factorizations for clustering. In: Eliassi-Rad T, Ungar LH, Craven M, Gunopulos D (eds) Proceedings of the Twelfth ACM SIGKDD Inter- national Conference on Knowledge Discovery and Data Mining, Philadelphia, PA, USA, ACM, pp 126–135 (2006), URL https://doi.org/10.1145/1150402.1150420
Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193–218 (1985). https://doi.org/10.1007/BF01908075
Kuhn, H.W.: The hungarian method for the assignment problem. Naval Res. Logistics quarerly 2(1–2), 83–97 (1955). https://doi.org/10.1002/nav.3800020109
Goldberger, J., Tassa, T.: A hierarchical clustering algorithm based on the hungarian method. Pattern Recognit. Lett. 29(11), 1632–1638 (2008). https://doi.org/10.1016/j.patrec.2008.04.003
Liang, N., Yang, Z., Li, Z., Xie, S., Su, C.: Semi-supervised multi-view clustering with graph-regularized partially shared non-negative matrix factorization. Knowl Based Syst 190, 105185 (2020). https://doi.org/10.1016/j.knosys.2019.105185
Peng, J., Luo, P., Guan, Z., Fan, J.: Graph-regularized multi-view semantic subspace learning. Int J Mach Learn Cybern 10(5), 879–895 (2019). https://doi.org/10.1007/s13042-017-0766-5
Acknowledgements
This work was supported in part by the Key-Area Research and Development Program of Guangdong Province under Grants 2019B010154002, 2019B010118001, and 2019B010121001; in part by the National Natural Science Foundation of China under Grants 61803096, 61801133, and U191140003; in part by the Guangzhou Science and Technology Program Project under Grant 202002030289; in part by the Guangdong Natural Science Foundation under Grant 2022A1515010688.
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Yang, Z., Zhang, H., Liang, N. et al. Semi-supervised multi-view clustering by label relaxation based non-negative matrix factorization. Vis Comput 39, 1409–1422 (2023). https://doi.org/10.1007/s00371-022-02419-z
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DOI: https://doi.org/10.1007/s00371-022-02419-z