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 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 highdimensional space and with an unknown number of subspaces are also presented. Our experiments on low-...

• Type: Conference
• Year: 2003
• Where: CVPR
• Authors: René Vidal, Shankar Sastry, Yi Ma
• Keywords: Computer Vision, CVPR 2003, Data Point, Sample Data Points, Subspace Segmentation, Temporal Video Segmentation, Unknown Number