Generalized principal component analysis (GPCA)
This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a...
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| Published in | IEEE transactions on pattern analysis and machine intelligence Vol. 27; no. 12; pp. 1945 - 1959 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Los Alamitos, CA
IEEE
01.12.2005
IEEE Computer Society The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects | |
| Online Access | Get full text |
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| Abstract | This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a high-dimensional space and with an unknown number of subspaces are also presented. Our experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k-subspaces and expectation maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views. |
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| AbstractList | When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a high-dimensional space and with an unknown number of subspaces are also presented. Our experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as K-subspaces and Expectation Maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views. This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a high-dimensional space and with an unknown number of subspaces are also presented. Our experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as K-subspaces and Expectation Maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views.This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a high-dimensional space and with an unknown number of subspaces are also presented. Our experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as K-subspaces and Expectation Maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views. |
| Author | Vidal, R. Sastry, S. Yi Ma |
| Author_xml | – sequence: 1 givenname: R. surname: Vidal fullname: Vidal, R. organization: Dept. of Biomed. Eng., Johns Hopkins Univ., Baltimore, MD, USA – sequence: 2 surname: Yi Ma fullname: Yi Ma – sequence: 3 givenname: S. surname: Sastry fullname: Sastry, S. |
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| CODEN | ITPIDJ |
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| Cites_doi | 10.1007/978-1-4684-9449-5 10.1016/S0031-3203(01)00198-4 10.1145/358669.358692 10.1109/CVPR.2003.1211482 10.1109/CVPR.2003.1211411 10.1109/CVPR.2004.1315075 10.1007/BF02288367 10.1137/S0036139998338583 10.1162/089976698300017467 10.1162/089976699300016728 10.1109/34.910882 10.1109/CVPR.1991.139704 10.1007/978-0-8176-4771-1 10.1007/978-1-4757-1904-8 10.1007/s11263-005-4839-7 10.1111/1467-9868.00196 10.1007/978-1-4757-2189-8 10.1007/978-1-4757-3849-0 10.1109/ICCV.2001.937679 10.1109/CVPR.2003.1211332 10.1023/A:1008000628999 |
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| Keywords | subspace segmentation Computer vision dynamic scenes and motion segmentation Veronese map temporal video segmentation Multiple view Index Terms-Principal component analysis (PCA) Image segmentation Dimension reduction dimensionality reduction Multidimensional database Scene analysis Tridimensional image EM algorithm Pattern analysis Principal component analysis |
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| References | ref13 hartshorne (ref8) 1977 vidal (ref26) 2004 ref15 harris (ref7) 1992 ref10 ref1 ref17 ref16 ref19 kanatani (ref14) 2003 vidal (ref22) 2004; ii jolliffe (ref12) 1986 wu (ref27) 2001; 2 ref24 stark (ref18) 2001 ref23 ref25 ref20 ref21 ref28 ref4 ref3 huang (ref11) 2004; 2 ref6 collins (ref2) 2001; 14 hirsch (ref9) 1976 ref5 |
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| SubjectTerms | Algorithms Application software Applied sciences Artificial Intelligence Cluster Analysis Clustering algorithms Computer science; control theory; systems Computer Simulation Computer vision Data points Derivatives dimensionality reduction dynamic scenes and motion segmentation Exact sciences and technology Image Enhancement - methods Image Interpretation, Computer-Assisted - methods Imaging, Three-Dimensional - methods Index Terms- Principal component analysis (PCA) Information Storage and Retrieval - methods Iterative algorithms Kernel Machine learning Mathematical analysis Mathematical models Matrix decomposition Models, Statistical Motion segmentation Pattern Recognition, Automated - methods Pattern recognition. Digital image processing. Computational geometry Polynomials Principal Component Analysis Principal components analysis Segmentation Studies subspace segmentation Subspaces temporal video segmentation Three dimensional Vectors (mathematics) Veronese map |
| Title | Generalized principal component analysis (GPCA) |
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