Generalized Principal Component Analysis (GPCA)

TL;DR

This paper introduces GPCA, an algebraic method for segmenting multiple subspaces of unknown dimensions from data, with applications in computer vision, outperforming existing algorithms and handling noise effectively.

Contribution

The paper develops a novel algebraic approach to subspace segmentation that handles unknown number and dimensions of subspaces, extending to high-dimensional data and noisy conditions.

Findings

GPCA outperforms existing algebraic algorithms in experiments.

It provides good initialization for iterative segmentation methods.

Applications include face clustering, video segmentation, and 3D motion segmentation.

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.