In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way.
Generalized Principal Component Analysis (GPCA) @cite_36 presents an algebraic way to model the data drawn from a union of multiple subspaces. This method describes a subspace containing a data point by using the gradient of a polynomial at that point. Then subspace segmentation is made equivalent to fitting the data with polynomials. GPCA can guarantee the success of the segmentation under certain conditions, and it does not impose any restriction on the subspaces. However, this method is sensitive to noise due to the difficulty of estimating the polynomials from real data, which also causes the high computation cost of GPCA. Recently, Robust Algebraic Segmentation (RAS) @cite_7 has been proposed to resolve the robustness issue of GPCA. However, the computation difficulty for fitting polynomials is unfathomably large. So RAS can make sense only when the data dimension is low and the number of subspaces is small.
@cite_36 @cite_7
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We propose an algebraic geometric approach to the problem of estimating a mixture of linear subspaces from sample data points, the so-called generalized principal component analysis (GPCA) problem. In the absence of noise, we show that GPCA is equivalent to factoring a homogeneous polynomial whose degree is the number of subspaces and whose factors (roots) represent normal vectors to each subspace. We derive a formula for the number of subspaces n and provide an analytic solution to the factorization problem using linear algebraic techniques. The solution is closed form if and only if n spl les 4. In the presence of noise, we cast GPCA as a constrained nonlinear least squares problem and derive an optimal function from which the subspaces can be directly recovered using standard nonlinear optimization techniques. We apply GPCA to the motion segmentation problem in computer vision, i.e. the problem of estimating a mixture of motion models from 2D imagery. 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. Recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high-dimensional data that have very complicated structures. From both geometric and statistical points of view, such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model. In this paper we propose to use subspace arrangements—a union of multiple subspaces—for modeling mixed data: each subspace in the arrangement is used to model just a homogeneous subset of the data. Thus, multiple subspaces together can capture the heterogeneous structures within the data set. In this paper, we give a comprehensive introduction to a new approach for the estimation of subspace arrangements. This is known as generalized principal component analysis (GPCA). In particular, we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust, even when the given data are corrupted by noise or contaminated with outliers. This new method in many ways improves and generalizes extant methods for modeling or clustering mixed data. There have been successful applications of this new method to many real-world problems in computer vision, image processing, and system identification. In this paper, we will examine several of those representative applications. This paper is intended to be expository in nature. However, in order that this may serve as a more complete reference for both theoreticians and practitioners, we take the liberty of filling in several gaps between the theory and the practice in the existing literature. We consider the problem of segmenting multiple rigid-body motions from point correspondences in multiple affine views. We cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two, three or four. Our algorithm involves projecting all point trajectories onto a 5-dimensional subspace using the SVD, the PowerFactorization method, or RANSAC, and fitting multiple linear subspaces representing different rigid-body motions to the points in ?5 using GPCA. Unlike previous work, our approach does not restrict the motion subspaces to be four-dimensional and independent. Instead, it deals gracefully with all the spectrum of possible affine motions: from two-dimensional and partially dependent to four-dimensional and fully independent. Our algorithm can handle the case of missing data, meaning that point tracks do not have to be visible in all images, by using the PowerFactorization method to project the data. In addition, our method can handle outlying trajectories by using RANSAC to perform the projection. We compare our approach to other methods on a database of 167 motion sequences with full motions, independent motions, degenerate motions, partially dependent motions, missing data, outliers, etc. On motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions.