Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

Thurau, Christian; Kersting, Kristian; Bauckhage, Christian

doi:10.1137/1.9781611972825.59

Cited by 12 publications

(9 citation statements)

References 30 publications

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“…Some approaches seek to maximize the volume of the decomposition [14, Figure 1: Comparison of singular vectors (left, scaled, in red) and DEIM-CUR columns (right, in blue) for a data set drawn from two multivariate normal distributions having different principal axes. 25]. Numerous other algorithms instead use leverage scores [5,10,22,28].…”

mentioning

confidence: 99%

A DEIM Induced CUR Factorization

Sorensen¹,

Embree²

2016

SIAM J. Sci. Comput.

112

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We derive a CUR approximate matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix A, such a factorization provides a low rank approximate decomposition of the form A ≈ CUR, where C and R are subsets of the columns and rows of A, and U is constructed to make CUR a good approximation. Given a low-rank singular value decomposition A ≈ VSW T , the DEIM procedure uses V and W to select the columns and rows of A that form C and R. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of V and W. For very large problems, V and W can be approximated well using an incremental QR algorithm that makes only one pass through A. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores. IntroductionThis work presents a new CUR matrix factorization based upon the Discrete Empirical Interpolation Method (DEIM). A CUR factorization is a low rank approximation of a matrix A ∈ R m×n of the form A ≈ CUR, where C = A(:, q) ∈ R m×k is a subset of the columns of A and R = A(p, :) ∈ R k×n is a subset of the rows of A.(We generally assume m ≥ n throughout.) The k × k matrix U is constructed to assure that CUR is a good approximation to A. Assuming the best rank-k singular value decomposition (SVD) A ≈ VSW T is available, the algorithm uses the DEIM index selection procedure, q = DEIM(V) and p = DEIM(W), to determine C and R. The resulting approximate factorization is nearly as accurate as the best rank-k SVD, withwhere σ k+1 is the first neglected singular value of A, η p ≡ V(p, : ) −1 , and η q ≡ W(q, : ) −1 .Here and throughout, · denotes the vector 2-norm and the matrix norm it induces, and · F is the Frobenius norm. We use MATLAB notation to index vectors and matrices, so that, e.g., A(p, :) denotes the k rows of A *

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mentioning

confidence: 99%

A DEIM Induced CUR Factorization

Sorensen¹,

Embree²

2016

SIAM J. Sci. Comput.

112

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“…The applicability of CUR decomposition in various fields can be found in [4,119,149]. The generalization of CUR decomposition to Tensors has been described in [21].…”

Section: Randomised Curmentioning

confidence: 99%

Literature survey on low rank approximation of matrices

Кумар

Schneider

2016

Linear and Multilinear Algebra

115

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Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic algorithms for low rank approximation. But these techniques are very expensive (O(n 3 ) operations are required for n × n matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n. In this article we review low rank approximation techniques briefly and give extensive references of many techniques.

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“…classifier based on statistical approach and classifier based on deterministic approach. Statistical approach [6] is used in a situation of availability for massive dataset while deterministic approach [7] is used for limited size of dataset. A good example of classifier based on statistical approach will be Support Vector Machine and Neural Network [8], while example for classifier for deterministic approach will be fuzzy logic and rule-based expert system [9].…”

Section: Figure 1 Standard Pq Classifier Designmentioning

confidence: 99%

Insight on Effectiveness of Frequently Exercised PQ Classification Techniques

Vighneshwari¹,

Neela²

2016

IJECE

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<p>The growing demands of global consumer market in green energy system have opened the doors for many technologies as well as various sophisticated electrical devices for both commercial and domestic usage. However, with the increasing demands of energy and better quality of services, there is a significant increase in non-linearity in load distribution causing potential effect on the Power Quality (PQ). The harmful effects on PQ are various events e.g. sag, swell, harmonics etc that causes significant amount of system degradation. Therefore, this paper discusses various significant research techniques pertaining to the PQ disturbance classification system introduced by the authors in the past and analyzes its effectiveness scale in terms of research gap. The paper discusses some of the frequently exercised PQ classification techniques from the most relevant literatures in order to have more insights of the techniques.</p>

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Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

Cited by 12 publications

References 30 publications

A DEIM Induced CUR Factorization

A DEIM Induced CUR Factorization

Literature survey on low rank approximation of matrices

Insight on Effectiveness of Frequently Exercised PQ Classification Techniques

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