Computing rank-revealing QR factorizations of dense matrices
ACM Transactions on Mathematical Software (TOMS)
Algorithm 782: codes for rank-revealing QR factorizations of dense matrices
ACM Transactions on Mathematical Software (TOMS)
Efficient Algorithms for the Block Hessenberg Form
The Journal of Supercomputing
On the Failure of Rank-Revealing QR Factorization Software -- A Case Study
ACM Transactions on Mathematical Software (TOMS)
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Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining
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Proceedings of the 17th ACM conference on Information and knowledge management
An improved approximation algorithm for the column subset selection problem
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
On selecting a maximum volume sub-matrix of a matrix and related problems
Theoretical Computer Science
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
SIAM Journal on Matrix Analysis and Applications
Column subset selection via sparse approximation of SVD
Theoretical Computer Science
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
An algorithm for the complete solution of quadratic eigenvalue problems
ACM Transactions on Mathematical Software (TOMS)
Improving CUR matrix decomposition and the Nyström approximation via adaptive sampling
The Journal of Machine Learning Research
Column Subset Selection Problem is UG-hard
Journal of Computer and System Sciences
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The problem of finding a rank-revealing QR (RRQR) factorisation of a matrix $A$ consists of permuting the columns of $A$ such that the resulting QR factorisation contains an upper triangular matrix whose linearly dependent columns are separated from the linearly independent ones. In this paper a systematic treatment of algorithms for determining RRQR factorisations is presented. In particular, the authors start by presenting precise mathematical formulations for the problem of determining a RRQR factorisation, all of them optimisation problems. Then a hierarchy of "greedy" algorithms is derived to solve these optimisation problems, and it is shown that the existing RRQR algorithms correspond to particular greedy algorithms in this hierarchy. Matrices on which the greedy algorithms, and therefore the existing RRQR algorithms, can fail arbitrarily badly are presented. Finally, motivated by an insight from the behaviour of the greedy algorithms, the authors present "hybrid" algorithms that solve the optimisation problems almost exactly (up to a factor proportional to the size of the matrix). Applying the hybrid algorithms as a follow-up to the conventional greedy algorithms may prove to be useful in practice.