基于信息几何的FSVM理论及算法研究
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摘要
支持向量机(SVM)是在统计学习理论上发展起来的一种机器学习方法。由于较好的解决了小样本、非线性、高维数、局部极小值等问题以及具有良好的推广能力,SVM已经成为机器学习领域的一个研究热点。但传统的支持向量机对样本中的噪声和孤立点非常敏感,为了克服这个问题,提出了模糊支持向量机(FSVM)理论。在模糊支持向量机中如何构造合适的隶属度函数成为FSVM首要解决的问题。另外,如何从几何角度改进FSVM也已成为当今研究FSVM的又一个热点。
本文详细论述了支持向量机理论、算法和性质,并通过构造合适的隶属度函数、核函数以及改进FSVM的几何结构,实现了对FSVM理论和算法的改进和完善。在对该课题进行深入研究后,主要做了如下创新性的工作:
(1)用从信息几何角度构造的动态核函数表示样本点和类中心的距离以及样本间的密切度,即把核方法的思想引入到距离和密切度的表示中;
(2)将基于类中心和密切度的模糊分类和回归隶属度函数分别进行组合,提出了基于类中心和密切度的乘积组合分类和回归隶属度函数,它们不仅考虑了样本点和类中心的距离,还考虑了样本间的密切度;
(3)从几何角度改进L-1范数FSVR的惩罚项,提出了L-2范数FSVR,仿真实验结果显示它比SVR和L-1范数FSVR具有更好的回归精度;
(4)提出了基于信息几何的模糊支持向量分类机和模糊支持向量回归机算法,并把基于信息几何的模糊支持向量分类机算法应用于图像边缘检测。
Support Vector Machine (SVM) is a machine learning method based on statistical learning theory.Because it resolves the small sample, nonlinearity, high dimension and local minimum problemsperfectly and has good generalization ability, SVM has been a hot field of machine learning.However, tradition SVM is very sensitive to noises and outliers in the training sample, in order toovercome this problem, the fuzzy support vector machine (FSVM) is proposed. It is how toconstruct a suitable membership function that has been the primary problem in FSVM. In addition,how to promote FSVM from the geometry aspect has became another hot topic.
This dissertation has elaborated the theory, algorithm and properties of SVM. The theory andalgorithm of FSVM perform more perfectly by constructing an appropriate membership function,kernel function and improving geometry of FSVM. After conducting a research deep into this topic,the main results are as follows:
(1) Using the dynamic kernel function, which is constructed from information geometry aspect,to express the distance between sample points and their cluster centers and the affinity among thesample points, that is to say, the kernel method is introduced to the representation of distance andaffinity;
(2) The product of combined fuzzy classification and regression membership function based onthe cluster center and the affinity are proposed. They are defined not only the distance between apoint and its cluster center, but also two different points of the sample, which is depicted as theaffinity between them;
(3) L-2 norm fuzzy support vector regression is proposed from the geometry aspect. Thesimulation results show that it has better regression accuracy than support vector regression and L-1norm fuzzy support vector regression;
(4) The algorithms of fuzzy support vector classification and fuzzy support vector regressionbased on information geometry are proposed. Then the fuzzy support vector classification algorithmis used to extract the edge of image.
本文详细论述了支持向量机理论、算法和性质,并通过构造合适的隶属度函数、核函数以及改进FSVM的几何结构,实现了对FSVM理论和算法的改进和完善。在对该课题进行深入研究后,主要做了如下创新性的工作:
(1)用从信息几何角度构造的动态核函数表示样本点和类中心的距离以及样本间的密切度,即把核方法的思想引入到距离和密切度的表示中;
(2)将基于类中心和密切度的模糊分类和回归隶属度函数分别进行组合,提出了基于类中心和密切度的乘积组合分类和回归隶属度函数,它们不仅考虑了样本点和类中心的距离,还考虑了样本间的密切度;
(3)从几何角度改进L-1范数FSVR的惩罚项,提出了L-2范数FSVR,仿真实验结果显示它比SVR和L-1范数FSVR具有更好的回归精度;
(4)提出了基于信息几何的模糊支持向量分类机和模糊支持向量回归机算法,并把基于信息几何的模糊支持向量分类机算法应用于图像边缘检测。
Support Vector Machine (SVM) is a machine learning method based on statistical learning theory.Because it resolves the small sample, nonlinearity, high dimension and local minimum problemsperfectly and has good generalization ability, SVM has been a hot field of machine learning.However, tradition SVM is very sensitive to noises and outliers in the training sample, in order toovercome this problem, the fuzzy support vector machine (FSVM) is proposed. It is how toconstruct a suitable membership function that has been the primary problem in FSVM. In addition,how to promote FSVM from the geometry aspect has became another hot topic.
This dissertation has elaborated the theory, algorithm and properties of SVM. The theory andalgorithm of FSVM perform more perfectly by constructing an appropriate membership function,kernel function and improving geometry of FSVM. After conducting a research deep into this topic,the main results are as follows:
(1) Using the dynamic kernel function, which is constructed from information geometry aspect,to express the distance between sample points and their cluster centers and the affinity among thesample points, that is to say, the kernel method is introduced to the representation of distance andaffinity;
(2) The product of combined fuzzy classification and regression membership function based onthe cluster center and the affinity are proposed. They are defined not only the distance between apoint and its cluster center, but also two different points of the sample, which is depicted as theaffinity between them;
(3) L-2 norm fuzzy support vector regression is proposed from the geometry aspect. Thesimulation results show that it has better regression accuracy than support vector regression and L-1norm fuzzy support vector regression;
(4) The algorithms of fuzzy support vector classification and fuzzy support vector regressionbased on information geometry are proposed. Then the fuzzy support vector classification algorithmis used to extract the edge of image.
引文
[1]邓乃扬,田英杰.数据挖掘中的新方法—支持向量机[M].北京:科学出版社,2004.
[2]邓乃扬,田英杰.支持向量机—理论、算法与拓展[M].北京:科学出版社,2009.
[3] Weston J. and Watkins C. Multi-class Support Vector Machines[C]. Proceedings of ESANN99,M.Verleysen, Brussels, Belgium, 1999.
[4] A. B. A. Graf, A.J. Smola, S. Borer. Classification in a normalized feature space using supportvector machines[J]. IEEE Transactions on Neural Networks, May 2003, 14(3): 597-605.
[5]王芬.基于SVM分类的图像边缘检测研究[D].宁夏:宁夏大学, 2010.
[6] Ying Tan, Jun Wang. A Support Vector Machine with a Hybrid Kernel and Minimal Vapnik-Chervonenkis Dimension[C]. IEEE Transactions on Knowledge and Data Engineering, 2004,16(4): 385-395.
[7]高隽.人工神经网络原理及仿真实例[M] .北京:机械工业出版社,2007.
[8] John S, Peter L. A framework for structural risk minimization[C]. In Proceedings, 9th AnnualConference on Computational Learning Theory, 1996:68~76.
[9] Burges CJC. A tutorial on support vector machines for pattern recognition[J]. Data Mining andKnowledge Discovery.1998, 2(2):121~167.
[10] Jianggui Cui, Zhonghai Li. The Application of Support Vector Machine in Pattern Recognition[C]. 2007 IEEE International Conference on Control and Automation, Guangzhou,2007:3135~3138.
[11]马潜云,张学工.支持向量机函数拟合在分形插值中的应用[J].清华大学学报(自然科学版),2000,40(3):76-78.
[12] Weston J, Gammerman A. Density estimation using support vector machines[C]. Technicalreport, Royal Holloway College, Report number CSD-TR-97-23,1997.
[13] C.F. Lin, S.D. Wang. Fuzzy support vector machines[J]. IEEE Transactions on NeuralNetworks, March 2002, 13(2): 464-471.
[14] Takuya Inoue, Shigeo Abe. Fuzzy Support Vector Machines for Pattern Classification[C].IEEE Proceedings of International Joint Conference on Neural Networks, July 2001: 1449-1454.
[15] H-P Huang,Y-H Liu. Fuzzy support vector machine for pattern recognition and data mining[J]. International Journal of Fuzzy Systems, 2002, 4(3): 826-835.
[16]张桂香,费岚,杜喆.基于类内超平面的模糊支持向量机[J].计算机工程与设计,2008,29(12):3177-3178,3207.
[17] Rao C.R. Inforamtion and Accuracy Attainable in the Estimation of Statistical Parameters,Bulletin of the Calcutta Mathematical Society, 1945,37:81-91.
[18] Chentsov N.N. Statistical Decision Rules and Optimal Inference, Rhode Island: AMS,1982.
[19] Efron B. The Geometry of Exponential Families. Annals of Statistics, 1978,6:326-376.
[20] Amari S. Differential-geometrical Methods in Statistics: Lecture Notes in Statistics,Springer-Verlag, Berlin Heidelberg New York Tokyo,1985.
[21]安金龙,王正欧,马振平.基于密度法的模糊支持向量机[J].天津大学学报,2004,37(6):544-548.
[22]张翔,肖小玲,徐光祐.基于样本之间紧密度的模糊支持向量机方法[J].软件学报,2006,17(5):951-958.
[23]吴青,刘三阳,杜喆.基于边界提取的模糊支持向量机方法[J].模式识别与人工智能,2008,21(3):332-337.
[24]祁立,刘玉树.基于两阶段聚类的模糊支持向量机[J].计算机工程,2008,34(1):4-6.
[25]祁立,刘玉树,基于聚类粒度的模糊支持向量机[J].哈尔滨工程大学学报, 2006,27(增刊):411-414.
[26] Amari. S, Kurata. K, Nagaoka. H. Information geometry of Boltzmann machines[J]. IEEEtransactions on neural networks, March 1992,3(2).
[27] Amaris, Wu si. An Information-geometrical method for improving the performance of supportvector machine classifiers [A]. Proceedings of International Conference on Artificial NeuralNetworks, 99 [C].London: IEEE Conference Publication,1999: 85-90.
[28] Amari, S.and Wu si. Improving Support Vector machine classifiers by modifying kernelfunction[J]. Neural Networks, 1999, 12:783-789.
[29] Streater R,F. Statistical Dynamics and Information Geometry. In H.Nencka and J.P.Bourguignon, editors, Geometry and Nature, Contemporary Mathematics, AmericanMathematical Society,1997,203:117-131.
[30] Grasselli M.R. Infinite Dimensional Quantum Information Geometry. In AIP ConferenceProceedings,2001,553(1):141-146.
[31] Amari S. Natural Gradient Works Efficiently in Learning, Neural Computation, 1998,10:251-276.
[32]李红英,钟波.支持向量分类机的核函数的研究[D].重庆:重庆大学,2009.
[33] Huan-jun Liu, Yao-nan Wang, Xiao-fen Lu. A Method to Choose Kernel Function and itsParameters for Support Vector Machines[C]. Proceedings of the Fourth InternationalConference on Machine Learning and Cybernetics, Guangzhou, 2005: 4277-4280.
[34]郭丽娟,孙世宇,段修生.支持向量机及核函数研究[J].科学技术与工程,2008,8(2):487-490.
[35]荣海娜,张葛祥,金炜东.系统辨识中支持向量机核函数及其参数的研究[J],系统仿真学报,2006,18(11):3204-3208.
[36] Chih-Cheng Yang, Wan-Jui Lee, and Shie-Jue Lee. Learning of Kernel Functions in SupportVector Machines[C]. 2006 International Joint Conference on Neural Networks SheratonVancouver Wall Centre Hotel, Vancouver, BC, Canada, 2006: 1150-1155.
[37] V. N. Vapnik. The Nature of Statistical Learning Theory [M]. Springer-Verlag, New York. 1995:5-13.
[38] V. N. Vapnik. Statistical learning theory [M]. Wiley, New York, 1998.
[39] V. N. Vapnik. An Overview of Statistical Learning Theory [J]. IEEE Transactions on NeuralNetworks, 1999, 10(5): 988-999.
[40] Vladimir N. Vapnik.统计学习理论的本质[M].张学工译.北京:清华大学出版社,2003.
[41]阎威武.支持向量机理论、方法和应用研究[D].上海:上海交通大学,2003.
[42] Cortes C, Vapnik V N. Supprot Vector networks[J]. Machine Learning, 1995,20:273-297.
[43]刘畅.模糊支持向量机[D].辽宁：辽宁师范大学,2008.
[44] Blumer A, Ehrenfeucht A, Haussler D et al. Learnability and the Vapnik-Chervonenkisdimension[J]. Journal of the ACM.1989, 36(4):929~965.
[45]孟媛媛.模糊支持向量机的研究与应用[D].山东:山东师范大学,2006.
[46]吴青.给予最优化理论的支持向量机学习算法研究[D].西安:西安电子科技大学,2006.
[47]苏金树,张博锋,徐昕.基于机器学习的文本分类研究进展[J].软件学报,2006,17(9):1848-1859.
[48]张成伟,郑诚.基于改进SVM的文本信息检索研究[J].计算机技术与发展,2009,19(1):71-73.
[49] Jia-ming Chen, Lei Li, Ling-ye Nie. Wavelet image compression by using hybrid kernelSVM[C]. Proceeding of ICMLC 2008, Kunming, July 2008: 3056-3060.
[50]王玉震,李雷.基于SVR的图像增强方法[J].计算机技术与发展,2009,19(1):60-62.
[51]徐彤阳,姚跃华,朱志勇.一种基于支持向量机的图像边缘检测方法[J].微机发展,2005,15(1): 87-90.
[52] M. Schmidt. Identifying speaker with support vector networks[C]. In Interface’96 Proceedings.Sydney,1996．
[53] S. Chakrabartty, GSingh, G. Cauwenberghs. Hybrid Support Vector Machine/Hidden MarkovModel approach for continuous speech recognition[C]. Proc.IEEE Midwest Symp Circuits andSystems(MWSCAS’2000),2000.
[54] P. Ramesh and J. G. Wilpon. Modeling state durations in hidden markov models for automaticspeech recognition[C]. In ICASSP-92,1992:381-384.
[55] Robert I. Damper, Steve R. Gunn, Mathew O. Gore. Extracting Phonetic Knowledge fromLearning Systems:Perce ptrons, Support Vector Machines and Linear Discriminants[J].AppliedIntelligence, 2000,12(1-2).
[56]张健沛,程丽丽,马骏.一种基于并行支持向量机的网络入侵检测方法[J].计算机工程与应用,2007,43(4):137-139.
[57] Ya-Li Ding, Lei Li, Hong-Qi Luo. A Novel Signature searching for intrusion detection SystemUsing Data Mining[C]. ICMLC, 2009: 122-126.
[58] Mukkamala S, Janoski G., Sung A. H. Intrusion Detection using Neural Networks and SupportVector Machines[C]. IEEE Proceedings on International Joint Conference on Neural Networks,2002.
[59]陈潇,李雷,范小岗.基于支持向量机的非线性多用户检测[J].西安邮电学院学报,2008, 13(1):37-40.
[60]李侃,高春晓,刘玉树.基于SVM的空间数据库的层次聚类分析[J].北京理工大学学报,2002,22(4):485-488.
[61]陈省身,陈维恒.微分几何讲义.北京:北京大学出版社,1980.
[62]刘蕴辉.基于信息几何的神经网络学习问题研究[D].北京:北京交通大学, 2005.
[63] WEN-SEN AN, YAN-GUANG SUN. An Information geometrical Approach to KernelConstruction in SVM and its Application in Soft Sensor Modeling[C], Proceeding of FourthInternational Conference on Machine Learning and Cybernetics,Guangzhou, August 2005:18-21.
[64]柳桂国,黄海燕,黄道.基于信息几何学概念的支持向量分类机[J].华东理工大学学报(自然科学版),2008, 34(3): 422-424.
[65]李雷,鲁延玲,周蒙蒙.基于核方法的一种新的模糊支持向量机[J].计算机技术与发展,2010,20(2): 9-15.
[66] Mengmeng Zhou, Lei Li and Yanling Lu, Fuzzy Support Vector Machine Based on Densitywith Dual Membership[C], IEEE Proceeding of ICMLC2009, Baoding, July 2009: 674-678.
[67]艾青,秦玉平,方辉,赵骥.一种扩展的紧密度模糊支持向量机及其在文本分类中应用[J].计算机应用与软件, 2010,26(4):45-47.
[68]孙名松,高庆国,王宣丹.基于双隶属度模糊支持向量机的邮件过滤[J].2010,46(2):93-95.
[69] Jiang Xiu-feng, Zhang Yi, Lv Jian-cheng. Fuzzy SVM with a new fuzzy membershipfunction[J].Neural Computation and Application, 2006,3(15):268-276.
[70]施其权,李小明,肖辞源.一类新型快速模糊支持向量机[J].计算机技术与发展,2010,20(2):103-105.
[71] Alireza Mansouri, Lilly Suriani Affendey, Ali Mamat. A New Fuzzy Support Vector MachineMethod for Named Entity Recognition[C]. International Conference on Computer Science andInformation Technology. 2008:24-28.
[72] Hong Xia. Feature Selection based on Fuzzy SVM[C] .Fifth International Conference on FuzzySystems and Knowledge Discovery, 2008:586-589.
[73] Kui Wu and Kim Hui Yap. Soft-Labeling Image Scheme Using Fuzzy Support VectorMachine[C]. studies in Computational Intelligence,2008: 96, 271-290.
[74]胡学坤,李金霞,宋淑娜,高尚.基于粗糙集与模糊支持向量机的模式分类方法研究[J].科技通报,2010, 26(2):249-252.
[75] The UCI repository of machine learning database, ftp://ftp.ics.uci.eud/pub/machine-learningdatabase.
[76]侯惠芳,白莉媛,刘素华.基于支持向量机的图像边缘检测研究[J].计算机工程与应用,2007,43(18):32-33.
[77]郑胜,柳健,田金文.基于向量机的边缘检测算法优化研究[J].电子与信息学报,2005,27(5):717-721.
[78]张文娟,贺兴时.基于组合核支持向量机的图像边缘检测[J].齐齐哈尔大学学报,2010,26(4):50-52.
[2]邓乃扬,田英杰.支持向量机—理论、算法与拓展[M].北京:科学出版社,2009.
[3] Weston J. and Watkins C. Multi-class Support Vector Machines[C]. Proceedings of ESANN99,M.Verleysen, Brussels, Belgium, 1999.
[4] A. B. A. Graf, A.J. Smola, S. Borer. Classification in a normalized feature space using supportvector machines[J]. IEEE Transactions on Neural Networks, May 2003, 14(3): 597-605.
[5]王芬.基于SVM分类的图像边缘检测研究[D].宁夏:宁夏大学, 2010.
[6] Ying Tan, Jun Wang. A Support Vector Machine with a Hybrid Kernel and Minimal Vapnik-Chervonenkis Dimension[C]. IEEE Transactions on Knowledge and Data Engineering, 2004,16(4): 385-395.
[7]高隽.人工神经网络原理及仿真实例[M] .北京:机械工业出版社,2007.
[8] John S, Peter L. A framework for structural risk minimization[C]. In Proceedings, 9th AnnualConference on Computational Learning Theory, 1996:68~76.
[9] Burges CJC. A tutorial on support vector machines for pattern recognition[J]. Data Mining andKnowledge Discovery.1998, 2(2):121~167.
[10] Jianggui Cui, Zhonghai Li. The Application of Support Vector Machine in Pattern Recognition[C]. 2007 IEEE International Conference on Control and Automation, Guangzhou,2007:3135~3138.
[11]马潜云,张学工.支持向量机函数拟合在分形插值中的应用[J].清华大学学报(自然科学版),2000,40(3):76-78.
[12] Weston J, Gammerman A. Density estimation using support vector machines[C]. Technicalreport, Royal Holloway College, Report number CSD-TR-97-23,1997.
[13] C.F. Lin, S.D. Wang. Fuzzy support vector machines[J]. IEEE Transactions on NeuralNetworks, March 2002, 13(2): 464-471.
[14] Takuya Inoue, Shigeo Abe. Fuzzy Support Vector Machines for Pattern Classification[C].IEEE Proceedings of International Joint Conference on Neural Networks, July 2001: 1449-1454.
[15] H-P Huang,Y-H Liu. Fuzzy support vector machine for pattern recognition and data mining[J]. International Journal of Fuzzy Systems, 2002, 4(3): 826-835.
[16]张桂香,费岚,杜喆.基于类内超平面的模糊支持向量机[J].计算机工程与设计,2008,29(12):3177-3178,3207.
[17] Rao C.R. Inforamtion and Accuracy Attainable in the Estimation of Statistical Parameters,Bulletin of the Calcutta Mathematical Society, 1945,37:81-91.
[18] Chentsov N.N. Statistical Decision Rules and Optimal Inference, Rhode Island: AMS,1982.
[19] Efron B. The Geometry of Exponential Families. Annals of Statistics, 1978,6:326-376.
[20] Amari S. Differential-geometrical Methods in Statistics: Lecture Notes in Statistics,Springer-Verlag, Berlin Heidelberg New York Tokyo,1985.
[21]安金龙,王正欧,马振平.基于密度法的模糊支持向量机[J].天津大学学报,2004,37(6):544-548.
[22]张翔,肖小玲,徐光祐.基于样本之间紧密度的模糊支持向量机方法[J].软件学报,2006,17(5):951-958.
[23]吴青,刘三阳,杜喆.基于边界提取的模糊支持向量机方法[J].模式识别与人工智能,2008,21(3):332-337.
[24]祁立,刘玉树.基于两阶段聚类的模糊支持向量机[J].计算机工程,2008,34(1):4-6.
[25]祁立,刘玉树,基于聚类粒度的模糊支持向量机[J].哈尔滨工程大学学报, 2006,27(增刊):411-414.
[26] Amari. S, Kurata. K, Nagaoka. H. Information geometry of Boltzmann machines[J]. IEEEtransactions on neural networks, March 1992,3(2).
[27] Amaris, Wu si. An Information-geometrical method for improving the performance of supportvector machine classifiers [A]. Proceedings of International Conference on Artificial NeuralNetworks, 99 [C].London: IEEE Conference Publication,1999: 85-90.
[28] Amari, S.and Wu si. Improving Support Vector machine classifiers by modifying kernelfunction[J]. Neural Networks, 1999, 12:783-789.
[29] Streater R,F. Statistical Dynamics and Information Geometry. In H.Nencka and J.P.Bourguignon, editors, Geometry and Nature, Contemporary Mathematics, AmericanMathematical Society,1997,203:117-131.
[30] Grasselli M.R. Infinite Dimensional Quantum Information Geometry. In AIP ConferenceProceedings,2001,553(1):141-146.
[31] Amari S. Natural Gradient Works Efficiently in Learning, Neural Computation, 1998,10:251-276.
[32]李红英,钟波.支持向量分类机的核函数的研究[D].重庆:重庆大学,2009.
[33] Huan-jun Liu, Yao-nan Wang, Xiao-fen Lu. A Method to Choose Kernel Function and itsParameters for Support Vector Machines[C]. Proceedings of the Fourth InternationalConference on Machine Learning and Cybernetics, Guangzhou, 2005: 4277-4280.
[34]郭丽娟,孙世宇,段修生.支持向量机及核函数研究[J].科学技术与工程,2008,8(2):487-490.
[35]荣海娜,张葛祥,金炜东.系统辨识中支持向量机核函数及其参数的研究[J],系统仿真学报,2006,18(11):3204-3208.
[36] Chih-Cheng Yang, Wan-Jui Lee, and Shie-Jue Lee. Learning of Kernel Functions in SupportVector Machines[C]. 2006 International Joint Conference on Neural Networks SheratonVancouver Wall Centre Hotel, Vancouver, BC, Canada, 2006: 1150-1155.
[37] V. N. Vapnik. The Nature of Statistical Learning Theory [M]. Springer-Verlag, New York. 1995:5-13.
[38] V. N. Vapnik. Statistical learning theory [M]. Wiley, New York, 1998.
[39] V. N. Vapnik. An Overview of Statistical Learning Theory [J]. IEEE Transactions on NeuralNetworks, 1999, 10(5): 988-999.
[40] Vladimir N. Vapnik.统计学习理论的本质[M].张学工译.北京:清华大学出版社,2003.
[41]阎威武.支持向量机理论、方法和应用研究[D].上海:上海交通大学,2003.
[42] Cortes C, Vapnik V N. Supprot Vector networks[J]. Machine Learning, 1995,20:273-297.
[43]刘畅.模糊支持向量机[D].辽宁：辽宁师范大学,2008.
[44] Blumer A, Ehrenfeucht A, Haussler D et al. Learnability and the Vapnik-Chervonenkisdimension[J]. Journal of the ACM.1989, 36(4):929~965.
[45]孟媛媛.模糊支持向量机的研究与应用[D].山东:山东师范大学,2006.
[46]吴青.给予最优化理论的支持向量机学习算法研究[D].西安:西安电子科技大学,2006.
[47]苏金树,张博锋,徐昕.基于机器学习的文本分类研究进展[J].软件学报,2006,17(9):1848-1859.
[48]张成伟,郑诚.基于改进SVM的文本信息检索研究[J].计算机技术与发展,2009,19(1):71-73.
[49] Jia-ming Chen, Lei Li, Ling-ye Nie. Wavelet image compression by using hybrid kernelSVM[C]. Proceeding of ICMLC 2008, Kunming, July 2008: 3056-3060.
[50]王玉震,李雷.基于SVR的图像增强方法[J].计算机技术与发展,2009,19(1):60-62.
[51]徐彤阳,姚跃华,朱志勇.一种基于支持向量机的图像边缘检测方法[J].微机发展,2005,15(1): 87-90.
[52] M. Schmidt. Identifying speaker with support vector networks[C]. In Interface’96 Proceedings.Sydney,1996．
[53] S. Chakrabartty, GSingh, G. Cauwenberghs. Hybrid Support Vector Machine/Hidden MarkovModel approach for continuous speech recognition[C]. Proc.IEEE Midwest Symp Circuits andSystems(MWSCAS’2000),2000.
[54] P. Ramesh and J. G. Wilpon. Modeling state durations in hidden markov models for automaticspeech recognition[C]. In ICASSP-92,1992:381-384.
[55] Robert I. Damper, Steve R. Gunn, Mathew O. Gore. Extracting Phonetic Knowledge fromLearning Systems:Perce ptrons, Support Vector Machines and Linear Discriminants[J].AppliedIntelligence, 2000,12(1-2).
[56]张健沛,程丽丽,马骏.一种基于并行支持向量机的网络入侵检测方法[J].计算机工程与应用,2007,43(4):137-139.
[57] Ya-Li Ding, Lei Li, Hong-Qi Luo. A Novel Signature searching for intrusion detection SystemUsing Data Mining[C]. ICMLC, 2009: 122-126.
[58] Mukkamala S, Janoski G., Sung A. H. Intrusion Detection using Neural Networks and SupportVector Machines[C]. IEEE Proceedings on International Joint Conference on Neural Networks,2002.
[59]陈潇,李雷,范小岗.基于支持向量机的非线性多用户检测[J].西安邮电学院学报,2008, 13(1):37-40.
[60]李侃,高春晓,刘玉树.基于SVM的空间数据库的层次聚类分析[J].北京理工大学学报,2002,22(4):485-488.
[61]陈省身,陈维恒.微分几何讲义.北京:北京大学出版社,1980.
[62]刘蕴辉.基于信息几何的神经网络学习问题研究[D].北京:北京交通大学, 2005.
[63] WEN-SEN AN, YAN-GUANG SUN. An Information geometrical Approach to KernelConstruction in SVM and its Application in Soft Sensor Modeling[C], Proceeding of FourthInternational Conference on Machine Learning and Cybernetics,Guangzhou, August 2005:18-21.
[64]柳桂国,黄海燕,黄道.基于信息几何学概念的支持向量分类机[J].华东理工大学学报(自然科学版),2008, 34(3): 422-424.
[65]李雷,鲁延玲,周蒙蒙.基于核方法的一种新的模糊支持向量机[J].计算机技术与发展,2010,20(2): 9-15.
[66] Mengmeng Zhou, Lei Li and Yanling Lu, Fuzzy Support Vector Machine Based on Densitywith Dual Membership[C], IEEE Proceeding of ICMLC2009, Baoding, July 2009: 674-678.
[67]艾青,秦玉平,方辉,赵骥.一种扩展的紧密度模糊支持向量机及其在文本分类中应用[J].计算机应用与软件, 2010,26(4):45-47.
[68]孙名松,高庆国,王宣丹.基于双隶属度模糊支持向量机的邮件过滤[J].2010,46(2):93-95.
[69] Jiang Xiu-feng, Zhang Yi, Lv Jian-cheng. Fuzzy SVM with a new fuzzy membershipfunction[J].Neural Computation and Application, 2006,3(15):268-276.
[70]施其权,李小明,肖辞源.一类新型快速模糊支持向量机[J].计算机技术与发展,2010,20(2):103-105.
[71] Alireza Mansouri, Lilly Suriani Affendey, Ali Mamat. A New Fuzzy Support Vector MachineMethod for Named Entity Recognition[C]. International Conference on Computer Science andInformation Technology. 2008:24-28.
[72] Hong Xia. Feature Selection based on Fuzzy SVM[C] .Fifth International Conference on FuzzySystems and Knowledge Discovery, 2008:586-589.
[73] Kui Wu and Kim Hui Yap. Soft-Labeling Image Scheme Using Fuzzy Support VectorMachine[C]. studies in Computational Intelligence,2008: 96, 271-290.
[74]胡学坤,李金霞,宋淑娜,高尚.基于粗糙集与模糊支持向量机的模式分类方法研究[J].科技通报,2010, 26(2):249-252.
[75] The UCI repository of machine learning database, ftp://ftp.ics.uci.eud/pub/machine-learningdatabase.
[76]侯惠芳,白莉媛,刘素华.基于支持向量机的图像边缘检测研究[J].计算机工程与应用,2007,43(18):32-33.
[77]郑胜,柳健,田金文.基于向量机的边缘检测算法优化研究[J].电子与信息学报,2005,27(5):717-721.
[78]张文娟,贺兴时.基于组合核支持向量机的图像边缘检测[J].齐齐哈尔大学学报,2010,26(4):50-52.