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Generalized Principal Component Analysis (GPCA)
December 2005 (vol. 27 no. 12)
pp. 1945-1959
ASCII Text
x 
 
Ren? Vidal, Yi Ma, Shankar Sastry, "Generalized Principal Component Analysis (GPCA)," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 12, pp. 1945-1959, December, 2005.
 
 
BibTex
x
 
@article{ 10.1109/TPAMI.2005.244,
author = {Ren? Vidal and Yi Ma and Shankar Sastry},
title = {Generalized Principal Component Analysis (GPCA)},
journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence},
volume = {27},
number = {12},
issn = {0162-8828},
year = {2005},
pages = {1945-1959},
doi = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.244},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Pattern Analysis and Machine Intelligence
TI - Generalized Principal Component Analysis (GPCA)
IS - 12
SN - 0162-8828
SP1945
EP1959
EPD - 1945-1959
A1 - Ren? Vidal,
A1 - Yi Ma,
A1 - Shankar Sastry,
PY - 2005
KW - Index Terms- Principal component analysis (PCA)
KW - subspace segmentation
KW - Veronese map
KW - dimensionality reduction
KW - temporal video segmentation
KW - dynamic scenes and motion segmentation.
VL - 27
JA - IEEE Transactions on Pattern Analysis and Machine Intelligence
ER -
 
Ren? Vidal, IEEE
Yi Ma, IEEE
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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Index Terms:
Index Terms- Principal component analysis (PCA), subspace segmentation, Veronese map, dimensionality reduction, temporal video segmentation, dynamic scenes and motion segmentation.
Citation:
Ren? Vidal, Yi Ma, Shankar Sastry, "Generalized Principal Component Analysis (GPCA)," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 12, pp. 1945-1959, Dec. 2005, doi:10.1109/TPAMI.2005.244
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