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FINDING STRUCTURE WITH RANDOMNESS: PROBABILISTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
"... Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for ..."
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Cited by 45 (0 self)
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Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing lowrank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired lowrank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition
FINDING STRUCTURE WITH RANDOMNESS: STOCHASTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
, 2009
"... Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing l ..."
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Cited by 27 (2 self)
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Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing lowrank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. In particular, these techniques offer a route toward principal component analysis (PCA) for petascale data. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired lowrank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider
IMPROVED ANALYSIS OF THE SUBSAMPLED RANDOMIZED HADAMARD TRANSFORM
"... Abstract. This paper presents an improved analysis of a structured dimensionreduction map called the subsampled randomized Hadamard transform. This argument demonstrates that the map preserves the Euclidean geometry of an entire subspace of vectors. The new proof is much simpler than previous appro ..."
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Cited by 11 (0 self)
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Abstract. This paper presents an improved analysis of a structured dimensionreduction map called the subsampled randomized Hadamard transform. This argument demonstrates that the map preserves the Euclidean geometry of an entire subspace of vectors. The new proof is much simpler than previous approaches, and it offers—for the first time—optimal constants in the estimate on the number of dimensions required for the embedding. 1.
Paved with good intentions: Analysis of a randomized Kaczmarz method
, 2012
"... ABSTRACT. The block Kaczmarz method is an iterative scheme for solving overdetermined leastsquares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized contro ..."
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Cited by 2 (0 self)
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ABSTRACT. The block Kaczmarz method is an iterative scheme for solving overdetermined leastsquares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving of the matrix, a partition of the rows into wellconditioned blocks. The operator theory literature provides detailed information about the existence and construction of good row pavings. Together, these results yield an efficient block Kaczmarz scheme that applies to many overdetermined leastsquares problem. 1.
Journal of Machine Learning Research () Submitted; Published Sampling Methods for the Nyström Method
"... The Nyström method is an efficient technique to generate lowrank matrix approximations and is used in several largescale learning applications. A key aspect of this method is the procedure according to which columns are sampled from the original matrix. In this work, we explore the efficacy of a v ..."
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The Nyström method is an efficient technique to generate lowrank matrix approximations and is used in several largescale learning applications. A key aspect of this method is the procedure according to which columns are sampled from the original matrix. In this work, we explore the efficacy of a variety of fixed and adaptive sampling schemes. We also propose a family of ensemblebased sampling algorithms for the Nyström method. We report results of extensive experiments that provide a detailed comparison of various fixed and adaptive sampling techniques, and demonstrate the performance improvement associated with the ensemble Nyström method when used in conjunction with either fixed or adaptive sampling schemes. Corroborating these empirical findings, we present a theoretical analysis of the Nyström method, providing novel error bounds guaranteeing a better convergence rate of the ensemble Nyström method in comparison to the standard Nyström method. 1.
Ensemble Nyström
"... A common problem in many areas of largescale machine learning involves manipulation of a large matrix. This matrix may be a kernel matrix arising in Support Vector Machines [9, 15], Kernel Principal Component Analysis [47] or manifold learning [43,51]. Large matrices also naturally arise in other a ..."
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A common problem in many areas of largescale machine learning involves manipulation of a large matrix. This matrix may be a kernel matrix arising in Support Vector Machines [9, 15], Kernel Principal Component Analysis [47] or manifold learning [43,51]. Large matrices also naturally arise in other applications, e.g., clustering, collaborative filtering, matrix completion, and robust PCA. For these largescale problems, the number of matrix entries can easily be in the order of billions or more, making them hard to process or even store. An attractive solution to this problem involves the Nyström method, in which one samples a small number of columns from the original matrix and generates its lowrank approximation using the sampled columns [53]. The accuracy of the Nyström method depends on the number columns sampled from the original matrix. Larger the number of samples, higher the accuracy but slower the method. In the Nyström method, one needs to perform SVD on a l × l matrix where l is the number of columns sampled from the original matrix. This SVD operation is typically carried out on a single machine. Thus, the maximum value of l used for an application is limited by the capacity of the machine. That is why in practice, one restricts l to be less than 20K or 30K, even when the size of matrix is in millions. This restricts the accuracy of the Nyström method in very largescale settings. This chapter describes a family of algorithms based on mixtures of Nyström approximations called, Ensemble Nyström algorithms, which yields more accurate lowrank approximations than the standard Nyström method. The core idea of Ensemble Nyström is to sample many subsets of columns from the original matrix, each containing a relatively small number of columns. Then, Nyström method is
Journal of Machine Learning Research 13 (2012) 9811006 Submitted 3/11; Revised 12/11; Published 4/12 Sampling Methods for the Nyström Method
"... The Nyström method is an efficient technique to generate lowrank matrix approximations and is used in several largescale learning applications. A key aspect of this method is the procedure according to which columns are sampled from the original matrix. In this work, we explore the efficacy of a v ..."
Abstract
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The Nyström method is an efficient technique to generate lowrank matrix approximations and is used in several largescale learning applications. A key aspect of this method is the procedure according to which columns are sampled from the original matrix. In this work, we explore the efficacy of a variety of fixed and adaptive sampling schemes. We also propose a family of ensemblebased sampling algorithms for the Nyström method. We report results of extensive experiments that provide a detailed comparison of various fixed and adaptive sampling techniques, and demonstrate the performance improvement associated with the ensemble Nyström method when used in conjunction with either fixed or adaptive sampling schemes. Corroborating these empirical findings, we present a theoretical analysis of the Nyström method, providing novel error bounds guaranteeing a better convergence rate of the ensemble Nyström method in comparison to the standard Nyström method.