Efficient algorithms for cur and interpolative matrix decompositions
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TL;DR: The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized library codes can be utilized for implementation.
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Abstract: The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized library codes can be utilized for implementation. For certain applications, further acceleration can be attained by incorporating techniques based on randomized projections. Numerical experiments demonstrate advantageous performance compared to existing techniques for computing CUR factorizations.
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Citations
Randomized numerical linear algebra: Foundations and algorithms
TL;DR: This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problems and treats both the theoretical foundations of the subject and practical computational issues.
Randomized Matrix Decompositions using R
TL;DR: This work presents the R package rsvd, and provides a tutorial introduction to randomized matrix decompositions, showing the computational advantage over other methods implemented in R for approximating matrices with low-rank structure.
Randomized Algorithms for Computation of Tucker Decomposition and Higher Order SVD (HOSVD)
Salman Ahmadi-Asl,Stanislav Abukhovich,Maame G. Asante-Mensah,Andrzej Cichocki,Anh Huy Phan,Toshihisa Tanaka,Ivan V. Oseledets +6 more
TL;DR: In this article, the authors review recent advances in randomization for computation of Tucker decomposition and Higher Order SVD (HOSVD) and discuss random projection and sampling approaches, single-pass and multi-pass randomized algorithms and how to utilize them in the computation of the Tucker decompposition and the HOSVD.
•Posted Content
Randomized Numerical Linear Algebra: Foundations & Algorithms.
TL;DR: This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problem instances and treats both the theoretical foundations and the practical computational issues.
55
Robust CUR Decomposition: Theory and Imaging Applications
TL;DR: In this article, the use of robust principal component analysis (RPCA) in a CUR decomposition framework and applications thereof is considered, and the main algorithms produce a robust version of column-row fact.
References
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
TL;DR: This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation, and presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions.
The approximation of one matrix by another of lower rank
Carl Eckart,Gale Young +1 more
TL;DR: In this paper, the problem of approximating one matrix by another of lower rank is formulated as a least-squares problem, and the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another.
4.2K
CUR matrix decompositions for improved data analysis
TL;DR: An algorithm is presented that preferentially chooses columns and rows that exhibit high “statistical leverage” and exert a disproportionately large “influence” on the best low-rank fit of the data matrix, obtaining improved relative-error and constant-factor approximation guarantees in worst-case analysis, as opposed to the much coarser additive-error guarantees of prior work.
Improved Approximation Algorithms for Large Matrices via Random Projections
Tamas Sarlos
- 21 Oct 2006
TL;DR: In this paper, the authors present a (1 + ∆)-approximation algorithm for the singular value decomposition of an m? n matrix A with M non-zero entries that requires 2 passes over the data and runs in time O(n 2 ).
1K
The Hat Matrix in Regression and ANOVA
David C. Hoaglin,Roy E. Welsch +1 more
TL;DR: A projection matrix known as the hat matrix contains this information and, together with the Studentized residuals, provides a means of identifying exceptional data points and simplifies the calculations involved in removing a data point.