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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We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the spl lscr sub 1 -norm. A number of recent theoretical results on sparsifying properties of spl lscr sub 1 penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits super-resolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Crame spl acute r-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization. Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rank-revealing QR , which seeks a well-conditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents a randomized, polynomial-time algorithm that produces the submatrix promised by Bourgain and Tzafriri. The method involves random sampling of columns, followed by a matrix factorization that exposes the well-conditioned subset of columns. This factorization, which is due to Grothendieck, is regarded as a central tool in modern functional analysis. The primary novelty in this work is an algorithm, based on eigenvalue minimization, for constructing the Grothendieck factorization. These ideas also result in a novel approximation algorithm for the @math norm of a matrix, which is generally NP -hard to compute exactly. As an added bonus, this work reveals a surprising connection between matrix factorization and the famous maxcut semidefinite program. Matrix factorization, when the matrix has missing values, has become one of the leading techniques for recommender systems. To handle web-scale datasets with millions of users and billions of ratings, scalability becomes an important issue. Alternating Least Squares (ALS) and Stochastic Gradient Descent (SGD) are two popular approaches to compute matrix factorization. There has been a recent flurry of activity to parallelize these algorithms. However, due to the cubic time complexity in the target rank, ALS is not scalable to large-scale datasets. On the other hand, SGD conducts efficient updates but usually suffers from slow convergence that is sensitive to the parameters. Coordinate descent, a classical optimization approach, has been used for many other large-scale problems, but its application to matrix factorization for recommender systems has not been explored thoroughly. In this paper, we show that coordinate descent based methods have a more efficient update rule compared to ALS, and are faster and have more stable convergence than SGD. We study different update sequences and propose the CCD++ algorithm, which updatesrank-one factors one by one. In addition, CCD++ can be easily parallelized on both multi-core and distributed systems. We empirically show that CCD++ is much faster than ALS and SGD in both settings. As an example, on a synthetic dataset with 2 billion ratings, CCD++ is 4 times faster than both SGD and ALS using a distributed system with 20 machines. Matrix factorization, when the matrix has missing values, has become one of the leading techniques for recommender systems. To handle web-scale datasets with millions of users and billions of ratings, scalability becomes an important issue. Alternating least squares (ALS) and stochastic gradient descent (SGD) are two popular approaches to compute matrix factorization, and there has been a recent flurry of activity to parallelize these algorithms. However, due to the cubic time complexity in the target rank, ALS is not scalable to large-scale datasets. On the other hand, SGD conducts efficient updates but usually suffers from slow convergence that is sensitive to the parameters. Coordinate descent, a classical optimization approach, has been used for many other large-scale problems, but its application to matrix factorization for recommender systems has not been thoroughly explored. In this paper, we show that coordinate descent-based methods have a more efficient update rule compared to ALS and have faster and more stable convergence than SGD. We study different update sequences and propose the CCD++ algorithm, which updates rank-one factors one by one. In addition, CCD++ can be easily parallelized on both multi-core and distributed systems. We empirically show that CCD++ is much faster than ALS and SGD in both settings. As an example, with a synthetic dataset containing 14.6 billion ratings, on a distributed memory cluster with 64 processors, to deliver the desired test RMSE, CCD++ is 49 times faster than SGD and 20 times faster than ALS. When the number of processors is increased to 256, CCD++ takes only 16 s and is still 40 times faster than SGD and 20 times faster than ALS.