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1、1000-9825/2005/16(08)1423©2005JournalofSoftware软件学报Vol.16,No.8∗高维数据流形的低维嵌入及嵌入维数研究1+121赵连伟,罗四维,赵艳敞,刘蕴辉1(北京交通大学计算机与信息技术学院,北京100044)2(FacultyofInformationTechnology,UniversityofTechnology,Sydney,Australia)StudyontheLow-DimensionalEmbeddingandtheEmbeddingDimensionalityofManifoldofHigh-Dimen
2、sionalData1+121ZHAOLian-Wei,LUOSi-Wei,ZHAOYan-Chang,LIUYun-Hui1(SchoolofComputerandInformationTechnology,BeijingJiaotongUniversity,Beijing100044,China)2(FacultyofInformationTechnology,UniversityofTechnology,Sydney,Australia)+Correspondingauthor:Phn:+86-10-51688556,E-mail:lw_zhao@hotmail
3、.com,http://www.bjtu.edu.cnReceived2004-07-14;Accepted2004-09-08ZhaoLW,LuoSW,ZhaoYC,LiuYH.Studyonthelow-dimensionalembeddingandtheembeddingdimensionalityofmanifoldofhigh-dimensionaldata.JournalofSoftware,2005,16(8):1423−1430.DOI:10.1360/jos161423Abstract:Findingmeaningfullow-dimensional
4、embeddedinahigh-dimensionalspaceisaclassicalproblem.Isomapisanonlineardimensionalityreductionmethodproposedandbasedonthetheoryofmanifold.Itnotonlycanrevealthemeaningfullow-dimensionalstructurehiddeninthehigh-dimensionalobservationdata,butcanrecovertheunderlyingparameterofdatalyingonalow
5、-dimensionalsubmanifold.Basedonthehypothesisthatthereisanisometricmappingbetweenthedataspaceandtheparameterspace,Isomapworks,butthishypothesishasnotbeenproved.Inthispaper,theexistenceofisometricmappingbetweenthemanifoldinthehigh-dimensionaldataspaceandtheparameterspaceisproved.Bydisting
6、uishingtheintrinsicdimensionalityofhigh-dimensionaldataspacefromthemanifolddimensionality,anditisprovedthattheintrinsicdimensionalityistheupperboundofthemanifolddimensionalityinthehigh-dimensionalspaceinwhichthereisatoroidalmanifold.Finallyanalgorithmisproposedtofindtheunderlyingtoroida
7、lmanifoldandjudgewhetherthereexistsone.Theresultsofexperimentsonthemulti-posethree-dimensionalobjectshowthatthemethodiseffective.Keywords:Isomap;toroidalmanifold;isometricmapping;embeddingdimensionality摘要:发现高维数据空间流形中有意义的低维嵌入是一个经典难题.Isomap是提出的一种有效的基于流形理论的非线性降维方法,它不仅能够揭示高维观察数据的内在
