Document Zbl 1444.62090

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Sci.</a> <a href="/?q=in%3A451036" title="Articles in this Issue">429, 247-259 (2018)</a>. </div> <div class="abstract">Summary: Least squares regression (LSR) is a widely used regression technique for multicategory classification. Conventional LSR model assumes that during the learning phase, the labeled samples can be exactly transformed into a discrete label matrix, which is too strict to learn a regression matrix for fitting the labels. To overcome this drawback, lots of LSR’s variants utilize the soft target label, which contains the continuous values, to replace this discrete label to improve the learning performance. Since the regression matrix can be learnt from these soft target labels, it is reasonable to assume that the samples in the same class have similar soft target labels. Nevertheless, most of existing LSR-based models don’t adequately consider this similarity assumption. In this paper, we propose a constrained least squares regression (CLSR) model for multicategory classification. The main motivation of CLSR is to force the samples in the same class to obtain the similar soft target labels. To effectively optimize CLSR, we propose a novel alternating algorithm, which can converge to the globally optimal solution. Extensive experiments results on face and digit data sets confirm the effectiveness of our proposed model.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7222567">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J02" title="MSC2020">62J02</a> </td> <td class="space"> General nonlinear regression </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62H30" title="MSC2020">62H30</a> </td> <td class="space"> Classification and discrimination; cluster analysis (statistical aspects) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aleast+squares+regression">least squares regression</a>; <a href="/?q=ut%3Asoft+target+label">soft target label</a>; <a href="/?q=ut%3Amulticategory+classification">multicategory classification</a></div> <div class="software"> <h3>Software:</h3><a href="/software/11003">aplore3</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1444.62090" data-ciurl="/ci/07222567" data-biburl="/bibtex/07222567.bib" data-amsurl="/amsrefs/07222567.bib" data-xmlurl="/xml/07222567.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07222567.pdf" title="Zbl 1444.62090 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.ins.2017.11.020" aria-label="DOI for “A constrained least squares regression model”" title="10.1016/j.ins.2017.11.020">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">An, S.; Liu, W.; Venkatesh, S., Face recognition using kernel ridge regression, Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit., 1-8 (2007)</td> </tr><tr> <td>[2]</td> <td class="space">Chang, K. W.; Hsieh, C. J.; Lin, C. J., Coordinate descent method for large-scale l2-loss linear support vector machines, J. Mach. Learn. Res., 9, 1369-1398 (2008) · <a href="/1225.68157" class="nowrap">Zbl 1225.68157</a></td> </tr><tr> <td>[3]</td> <td class="space">Cortes, C.; Vapnik, V., Support-vector networks, Mach. Learn., 20, 3, 273-297 (1995) · <a href="/0831.68098" class="nowrap">Zbl 0831.68098</a></td> </tr><tr> <td>[4]</td> <td class="space">Crammer, K.; Singer, Y., On the algorithmic implementation of multiclass kernel-based vector machines, J. Mach. Learn. Res., 2, 265-292 (2002) · <a href="/1037.68110" class="nowrap">Zbl 1037.68110</a></td> </tr><tr> <td>[5]</td> <td class="space">Cui, C.; Wang, D., High dimensional data regression using lasso model and neural networks with random weights, Inf. Sci., 372, 505-517 (2016)</td> </tr><tr> <td>[6]</td> <td class="space">Fang, X.; Xu, Y.; Li, X.; Lai, Z.; Wong, W. K.; Fang, B., Regularized label relaxation linear regression, IEEE Trans. Neural Netw. Learn. Syst. (2017)</td> </tr><tr> <td>[7]</td> <td class="space">Feng, P.; Dang, C., Shape constrained risk-neutral density estimation by support vector regression, Inf. Sci., 333, 1-9 (2016) · <a href="/1395.62078" class="nowrap">Zbl 1395.62078</a></td> </tr><tr> <td>[8]</td> <td class="space">Georghiades, A.; Belhumeur, P.; Kriegman, D., From few to many: illumination cone models for face recognition under variable lighting and pose, IEEE Trans. Pattern Anal. Mach. Intell., 23, 6, 643-660 (2001)</td> </tr><tr> <td>[9]</td> <td class="space">Gui, J.; Sun, Z.; Ji, S.; Tao, D.; Tan, T., Feature selection based on structured sparsity: a comprehensive study, IEEE Trans. Neural Netw. Learn. Syst., 28, 7, 1490-1507 (2017)</td> </tr><tr> <td>[10]</td> <td class="space">Hoerl, A.; Kennard, R., Ridge regression: biased estimation for nonorthogonal problems, Technometrics, 12, 1, 55-67 (1970) · <a href="/0202.17205" class="nowrap">Zbl 0202.17205</a></td> </tr><tr> <td>[11]</td> <td class="space">Hosmer, D. W.; Lemeshow, J. S.; Sturdivant, R. X., Applied Logistic Regression (2013), Wiley: Wiley New York, NY, USA · <a href="/1276.62050" class="nowrap">Zbl 1276.62050</a></td> </tr><tr> <td>[12]</td> <td class="space">Hou, C.; Jiao, Y.; Nie, F.; Luo, T.; Zhou, Z. H., 2D feature selection by sparse matrix regression, IEEE Trans. Image Process., 26, 9, 4255-4268 (2017) · <a href="/1409.94230" class="nowrap">Zbl 1409.94230</a></td> </tr><tr> <td>[13]</td> <td class="space">Hsu, C.-W.; Lin, C.-J., A comparison of methods for multiclass support vector machines, IEEE Trans. Neural Netw., 13, 2, 415-425 (2002)</td> </tr><tr> <td>[14]</td> <td class="space">Hull, J., A database for handwritten text recognition research, IEEE Trans. Pattern Anal. Mach. Intell., 16, 5, 550-554 (1994)</td> </tr><tr> <td>[15]</td> <td class="space">Krishnapuram, B.; Carin, L.; Figueiredo, M. A.T.; Hartemink, A. J., Sparse multinomial logistic regression: fast algorithms and generalization bounds, IEEE Trans. Pattern Anal. Mach. Intell., 27, 6, 957-968 (2005)</td> </tr><tr> <td>[16]</td> <td class="space">Ma, J.; Chow, T. W., Robust non-negative sparse graph for semi-supervised multi-label learning with missing labels, Inf. Sci., 422, 336-351 (2018) · <a href="/1436.68311" class="nowrap">Zbl 1436.68311</a></td> </tr><tr> <td>[17]</td> <td class="space">Martinez, A.; Benavente, R., A database for handwritten text recognition research, CVC Technical Report 24 (1998)</td> </tr><tr> <td>[18]</td> <td class="space">Melki, G.; Cano, A.; Kecman, V.; Ventura, S., Multi-target support vector regression via correlation regressor chains, Inf. Sci., 415, 53-69 (2017) · <a href="/1443.62160" class="nowrap">Zbl 1443.62160</a></td> </tr><tr> <td>[19]</td> <td class="space">Naseem, I.; Togneri, R.; Bennamoun, M., Linear regression for face recognition, IEEE Trans. Pattern Anal. Mach. Intell., 32, 11, 2106-2112 (2010)</td> </tr><tr> <td>[20]</td> <td class="space">Peng, X.; Shen, J., A twin-hyperspheres support vector machine with automatic variable weights for data classification, Inf. Sci., 417, 216-235 (2017) · <a href="/1435.68281" class="nowrap">Zbl 1435.68281</a></td> </tr><tr> <td>[21]</td> <td class="space">Samaria, F.; Harter, A., Parameterisation of a stochastic model for human face identification, Proc. 2nd IEEE Workshop Applications of Computer Vision, Sarasota, FL, 138-142 (1994)</td> </tr><tr> <td>[22]</td> <td class="space">Sidahmed, H.; Prokofyeva, E.; Blaschko, M. B., Discovering predictors of mental health service utilization with k-support regularized logistic regression, Inf. Sci., 329, 937-949 (2016)</td> </tr><tr> <td>[23]</td> <td class="space">Tao, H.; Hou, C.; Nie, F.; Zhu, J.; Yi, D., Scalable multi-view semi-supervised classification via adaptive regression, IEEE Trans. Image Process., 26, 9, 4283-4296 (2017) · <a href="/1409.94583" class="nowrap">Zbl 1409.94583</a></td> </tr><tr> <td>[24]</td> <td class="space">Tibshirani, R., Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. B, 58, 1, 267-288 (1996) · <a href="/0850.62538" class="nowrap">Zbl 0850.62538</a></td> </tr><tr> <td>[25]</td> <td class="space">Wang, L.; Pan, C., Groupwise retargeted least-squares regression, IEEE Trans. Neural Netw. Learn. Syst. (2017)</td> </tr><tr> <td>[26]</td> <td class="space">Wright, J.; Yang, A.; Ganesh, A.; Sastry, S.; Ma, Y., Robust face recognition via sparse representation, IEEE Trans. Pattern Anal. Mach. Intell., 31, 2, 210-227 (2009)</td> </tr><tr> <td>[27]</td> <td class="space">Xiang, S.; Nie, F.; Meng, G.; Pan, C.; Zhang, C., Discriminative least squares regression for multiclass classification and feature selection, IEEE Trans. Neural Netw. Learn. Syst., 23, 11, 1738-1754 (2012)</td> </tr><tr> <td>[28]</td> <td class="space">Xu, C.; Liu, T.; Tao, D.; Xu, C., Local rademacher complexity for multi-label learning, IEEE Trans. Image Process., 25, 3, 1495-1507 (2016) · <a href="/1408.94734" class="nowrap">Zbl 1408.94734</a></td> </tr><tr> <td>[29]</td> <td class="space">Xu, C.; Tao, D.; Xu, C., Multi-view learning with incomplete views, IEEE Trans. Image Process., 24, 12, 5812-5825 (2015) · <a href="/1408.94735" class="nowrap">Zbl 1408.94735</a></td> </tr><tr> <td>[30]</td> <td class="space">Xu, C.; Tao, D.; Xu, C., Robust extreme multi-label learning, Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 1275-1284 (2016)</td> </tr><tr> <td>[31]</td> <td class="space">Zhang, X. Y.; Wang, L.; Xiang, S.; Liu, C. L., Retargeted least squares regression algorithm, IEEE Trans. Neural Netw. Learn. Syst., 26, 9, 2206-2213 (2015)</td> </tr><tr> <td>[32]</td> <td class="space">Zhang, Z.; Lai, Z.; Xu, Y.; Shao, L.; Wu, J.; Xie, G. S., Discriminative elastic-net regularized linear regression, IEEE Trans. Image Process., 26, 3, 1466-1481 (2017) · <a href="/1409.94783" class="nowrap">Zbl 1409.94783</a></td> </tr><tr> <td>[33]</td> <td class="space">Zhang, Z.; Zhai, Z.; Li, L., Uniform projection for multi-view learning, IEEE Trans. Pattern Anal. Mach. Intell., 39, 8, 1675-1689 (2017)</td> </tr><tr> <td>[34]</td> <td class="space">Zhen, X.; Yu, M.; He, X.; Li, S., Multi-target regression via robust low-rank learning, IEEE Trans. Pattern Anal. Mach. Intell. (2017)</td> </tr><tr> <td>[35]</td> <td class="space">Zhen, X.; Yu, M.; Zheng, F.; Nachum, I. B.; Bhaduri, M.; Laidley, D.; Li, S., Multitarget sparse latent regression, IEEE Trans. Neural Netw. Learn. Syst. (2017)</td> </tr><tr> <td>[36]</td> <td class="space">Zhu, X.; Li, X.; Zhang, S.; Ju, C.; Wu, X., Robust joint graph sparse coding for unsupervised spectral feature selection, IEEE Trans. Neural Netw. Learn. Syst., 28, 6, 1263-1275 (2017)</td> </tr><tr> <td>[37]</td> <td class="space">Zou, J.; Li, W.; Chen, C.; Du, Q., Scene classification using local and global features with collaborative representation fusion, Inf. Sci., 348, 209-226 (2016)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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