Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (3): 1173-1190.doi: 10.1007/s10473-022-0321-7
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Shuhua WANG1, Baohuai SHENG2,3
Received:
2020-12-24
Revised:
2021-04-20
Online:
2022-06-26
Published:
2022-06-24
Contact:
Baohuai SHENG,E-mail:shengbaohuai@163.com
E-mail:shengbaohuai@163.com
Supported by:
CLC Number:
Shuhua WANG, Baohuai SHENG. LEARNING RATES OF KERNEL-BASED ROBUST CLASSIFICATION[J].Acta mathematica scientia,Series A, 2022, 42(3): 1173-1190.
[1] | Chen D R, Wu Q, Ying Y M, Zhou D X. Support vector machine soft margin classifier:error analysis[J]. J Mach Learn Res, 2004, 5:1143-1175 |
[2] | Xiang D H, Zhou D X. Classification with Guassians and convex loss[J]. J Mach Learn Res, 2009, 10:1447-1468 |
[3] | Cao F L, Dai T H, Zhang Y Q. Generalization bounds of compressed regression learning algorithm[J]. Acta Mathematica Scientia, 2014, 34A(4):905-916 |
[4] | Zhou D X, Jetter K. Approximation with polynomial kernels and SVM classifiers[J]. Adv Comput Math, 2006, 25:323-344 |
[5] | Sheng B H, Xiang D H. The learning rate of l2-coefficient regularized classification with strong loss[J]. Acta Mathematica Sinica, 2013, 29B(12):2397-2408 |
[6] | Xiang D H. Classification with Gaussians and convex loss II:Improving error bounds by noise conditions[J]. Science in China:Mathematics, 2011, 54(10):165-171 |
[7] | Tikhonov A, Arsenin V. Solutions of ill-posed problems[M]. W H Winston, 1977 |
[8] | Hou Z Y, Yang H Q. Regularization method for improving optimal convergence rate of the regularized solution of ill-posed problems[J]. Acta Mathematica Scientia, 1998, 18B(2):177-185 |
[9] | Chen H. On the asymptotic order of Tikhonov regularization with operators and data error[J]. Acta Mathematica Scientia, 1998, 18B(1):35-44 |
[10] | Pang Meng J, Sun H W. Distributed learning with partial coefficients regularization[J/OL]. Int J Wavelets Multiresolut Inf Process, 2018, 29(04):1850025. https://doi.org/10.1142/S021969131850025X |
[11] | Wu Q, Ying Y M, Zhou D X. Multi-kernel regularized classifiers[J]. J Complexity, 2007, 23:108-134 |
[12] | Zhang T. Statistical behavior and consistency of classification methods based on convex risk minimization[J]. Annals of Statistics, 2004, 32:56-85 |
[13] | Zhang J, Wang J L, Sheng B H. Learning from regularized regression algorithms with p-order Markov chain sampling[J]. Appl Math J Chinese Univ, 2011, 26B(3):295-306 |
[14] | Tong H Z, Chen D R, Peng L Z. Learning rates for regularized classifiers using multivariate polynomial kernels[J]. J Complexity, 2008, 24:619-631 |
[15] | Rosasco L, De VitoE, Caponnetto A, Piana M. Are loss functions all the same?[J]. Neural Computation, 2014, 16(5):1063-1076 |
[16] | Cristianini N, Shawe-Taylor J. An Introduction to Support Vector Machines[M]. Cambridge University Press, 2000 |
[17] | Steinwart I, Christman A. Support vector machines[M/OL]. Springer, 2008. https://doi.org/10.1007/978-0-387-77242-4 |
[18] | Sun H W, Liu P. Regularized least square algorithm with two kernels[J]. Int J Wavelets Multiresolut Inf Process, 2012, 10(5):1250043. Doi:https://doi.org/10.1142/S0219691312500439 |
[19] | Sun H W, Liu P. The optimal solution of multi-kernel regularization learning[J]. Acta Mathematica Sinica, English Series, 2013, 29(8):1607-1616 |
[20] | Zhao Z H, Lin Y Z. Reproducing kernel method for piecewise smooth boundary value problems[J]. Acta Mathematica Scientia, 2020, 40A(5):1333-1340 |
[21] | Sheng B H. The weighted norm for some Mercer kernel matrices[J]. Acta Mathematica Scientia, 2013, 33A(1):6-15 |
[22] | Sheng B H, Zhang H Z. Performance analysis of the LapRSSLG algorithm in learning theory[J]. Anal Appl, 2020, 18(1):79-108 |
[23] | Rachdi L T, Msehli N. Best approximation for weierstrass transform connected with spherical mean operator[J]. Acta Mathematica Scientia, 2012, 32B(2):455-470 |
[24] | Aggarwal Charu C. Outlier Analysis[M]. Springer, 2017 |
[25] | Hampel F R, Ronchetti E M, Rousseeuw P J, Stahel W A. Robust statistics:the approach based on influence functions. New York:John Wiley Sons, 1986 |
[26] | Tsyurmasto P, Zabarankin M. Value-at-risk support vector machine:stability to outliers[J]. J Comb Optim, 2014, 28:218-232 |
[27] | Huber P, Ronchetti E. Robust Statistics[M]. 2nd edition. Wiley, 2009 |
[28] | Wu Y, Liu Y. Robust truncated hinge loss support vector machine[J]. J American Statistical Association, 2007, 102:974-983 |
[29] | Freund Y. A more robust boosting algorithm[J/OL]. Ann Statist, May 2009. Preprint arXiv:0905.2138. http://arxiv.org/abs/0905.2138 |
[30] | Masnadi-Shiraze H, Vasconcelos N. On the design of loss functions for classification:theory, robustness to outliers, and savegeboost[J]. In Advances in Neural Information Processing Systems, 2009, 22:1049-1056 |
[31] | Suzumura S, Ogawa K, Sugiyama, Karasuyama M, Takeuchi I. Homotopy continuation approaches for robust SV classification and regression[J]. Mach Learn, 2017, 106:1009-1038 |
[32] | Sheng B H, Liu H X, Wang H M. Learning rates for the kernel regularized regression with a differentiable stongly convex loss[J]. Commun Pur Appl Anal, 2020, 19(8):3973-4005 |
[33] | Sheng B H, Wang J L. On the K-functional in learning theory[J]. Anal Appl, 2020, 18(3):423-446 |
[34] | Sheng B H, Wang J L, Xiang D H. Error analysis on Hérmite learning with gradient data[J]. Chin Ann Math Ser B, 2018, 39(4):705-720 |
[35] | Wang S H, Chen Z L, Sheng B H. Convergence of online pairwise regression learning with quadratic loss[J]. Commun Pur Appl Anal, 2020, 19(8):4023-4054 |
[36] | Sheng B H, Zuo L. Error analysis of the kernel regularized regression based on refined convex losses and RKBSs[J/OL]. Int J Wavelets Multiresolut Inf Process, 2021:2150012 (52 pages). https://doi.org/10.1142/S0219691321500120 |
[37] | Bartlett P L, Jordan M I, McAuliffe J D. Convex, classification, and risk bounds[J]. Journal of American Statistical Association, 2006, 101:138-156 |
[38] | Wang S H, Wang Y J, Chen Z L, Sheng B H. The convergence rate for kernel-based regularized pair learning algorithm with a quasiconvex loss[J]. J Sys Sci & Math Scis, 2020, 40(3):389-409 (in Chinese) |
[39] | Wang S H, Chen Z L, Sheng B H. The convergence rate of SVM for kernel-based robust regression[J/OL]. Int J Wavelets Multiresolut Inf Process, 2019, 17(1):1950004 (21 pages). https://doi.org/10.1142/S0219691319500048 |
[40] | Cambini A, Martein L. Generalized convexity and optimization:theory and applications[M]. Springerverlag, 2009 |
[41] | Micchelli C A, Xu Y S, Zhang H Z. Universal kernels[J]. J Mach Learn Res, 2006, 7:2651-2667 |
[42] | Smale S, Zhou D X. Estimating the approximation error in learning theory[J]. Anal and Appli, 2003, 1(1):1-25 |
[43] | Smale S, Zhou D X. Learning theory estimates via integral operators and their applications[J]. Constr Approx, 2007, 26(2):153-172 |
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