This paper describes a new approach, based on linear programming, for computing nonnegative matrix factorizations (NMFs). The key idea is a data-driven model for the factorization where the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identifies a matrix C such that X approximately equals CX and some linear constraints. The constraints are chosen to ensure that the matrix C selects features; these features can then be used to find a low-rank NMF of X. A theoretical analysis demonstrates that this approach has guarantees similar to those of the recent NMF algorithm of (2012). In contrast with this earlier work, the proposed method extends to more general noise models and leads to efficient, scalable algorithms. Experiments with synthetic and real datasets provide evidence that the new approach is also superior in practice. An optimized C++ implementation can factor a multigigabyte matrix in a matter of minutes.
Localizing factorizations via column or row subset selection is a popular alternative to direct factorization methods such as the SVD. Interpolative decomposition such as Rank-Revealing QR @cite_7 and CUR @cite_25 have favorable efficiency properties as compared to factorizations (such as SVD) that are not based on exemplars. Factorization localization has been used in subspace clustering and has been shown to be robust to outliers @cite_22 @cite_19 .
@cite_19 @cite_22 @cite_25 @cite_7
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This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [11], which signicantly broadens the range of problems where it is provably eective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also prove that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. A numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods. We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that, under mild assumptions, the SR can be obtained exactly' by using l1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR. Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that our approach significantly outperforms state-of-the-art methods. Principal components analysis and, more generally, the Singular Value Decomposition are fundamental data analysis tools that express a data matrix in terms of a sequence of orthogonal or uncorrelated vectors of decreasing importance. Unfortunately, being linear combinations of up to all the data points, these vectors are notoriously difficult to interpret in terms of the data and processes generating the data. In this article, we develop CUR matrix decompositions for improved data analysis. CUR decompositions are low-rank matrix decompositions that are explicitly expressed in terms of a small number of actual columns and or actual rows of the data matrix. Because they are constructed from actual data elements, CUR decompositions are interpretable by practitioners of the field from which the data are drawn (to the extent that the original data are). We present an algorithm that preferentially chooses columns and rows that exhibit high “statistical leverage” and, thus, in a very precise statistical sense, exert a disproportionately large “influence” on the best low-rank fit of the data matrix. By selecting columns and rows in this manner, we obtain improved relative-error and constant-factor approximation guarantees in worst-case analysis, as opposed to the much coarser additive-error guarantees of prior work. In addition, since the construction involves computing quantities with a natural and widely studied statistical interpretation, we can leverage ideas from diagnostic regression analysis to employ these matrix decompositions for exploratory data analysis. Given anm n matrixM withm > n, it is shown that there exists a permutation FI and an integer k such that the QR factorization MYI= Q(Ak ckBk) reveals the numerical rank of M: the k k upper-triangular matrix Ak is well conditioned, IlCkll2 is small, and Bk is linearly dependent on Ak with coefficients bounded by a low-degree polynomial in n. Existing rank-revealing QR (RRQR) algorithms are related to such factorizations and two algorithms are presented for computing them. The new algorithms are nearly as efficient as QR with column pivoting for most problems and take O (ran2) floating-point operations in the worst case.