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1、1000-9825/2005/16(08)1423?2005JournalofSoftware軟件學報Vol.16,No.8?高維數據流形的低維嵌入及嵌入維數研究1+121趙連偉,羅四維,趙艷敞,劉蘊輝1(北京交通大學計算機與信息技術學院,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:發(fā)現高維數據空間流形中有意義的低維嵌入是一個經典難題.Isomap是提出的一種有效的基于流形理論的非線性降維方法,它不僅能夠揭示高維觀察數據的內在
