Sensitivity analysis in linear regression
Sensitivity analysis in linear regression
Some applications of the rank revealing QR factorization
SIAM Journal on Scientific and Statistical Computing
On Rank-Revealing Factorisations
SIAM Journal on Matrix Analysis and Applications
Efficient algorithms for computing a strong rank-revealing QR factorization
SIAM Journal on Scientific Computing
Computing rank-revealing QR factorizations of dense matrices
ACM Transactions on Mathematical Software (TOMS)
Competitive recommendation systems
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Fast Monte-Carlo Algorithms for finding low-rank approximations
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Ranking a random feature for variable and feature selection
The Journal of Machine Learning Research
Matrix approximation and projective clustering via volume sampling
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Tensor-CUR decompositions for tensor-based data
Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining
The Journal of Machine Learning Research
Sampling from large matrices: An approach through geometric functional analysis
Journal of the ACM (JACM)
Spectral feature selection for supervised and unsupervised learning
Proceedings of the 24th international conference on Machine learning
Unsupervised feature selection for principal components analysis
Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining
Relative-Error $CUR$ Matrix Decompositions
SIAM Journal on Matrix Analysis and Applications
Tensor-CUR Decompositions for Tensor-Based Data
SIAM Journal on Matrix Analysis and Applications
Adaptive sampling and fast low-rank matrix approximation
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Subspace sampling and relative-error matrix approximation: column-based methods
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Identifying critical variables of principal components for unsupervised feature selection
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
Low rank matrix-valued chernoff bounds and approximate matrix multiplication
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Column subset selection via sparse approximation of SVD
Theoretical Computer Science
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
Sampling methods for the Nyström method
The Journal of Machine Learning Research
Simple and deterministic matrix sketching
Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining
Sparse and unique nonnegative matrix factorization through data preprocessing
The Journal of Machine Learning Research
Fast approximation of matrix coherence and statistical leverage
The Journal of Machine Learning Research
A scalable approach to column-based low-rank matrix approximation
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
A note on sparse least-squares regression
Information Processing Letters
Column Subset Selection Problem is UG-hard
Journal of Computer and System Sciences
| Hi-index | 0.02 |
We consider the problem of selecting the "best" subset of exactly k columns from an m x n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn2, m2n}) time and returns as output an m x k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-chosen probability distribution that depends on information in the top-k right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic column-selection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m x k matrix containing those k columns, let PC denote the projection matrix onto the span of those columns, and let Ak denote the "best" rank-k approximation to the matrix A as computed with the singular value decomposition. Then, we prove that [EQUATION] with probability at least 0.7. This spectral norm bound improves upon the best previously-existing result (of Gu and Eisenstat [21]) for the spectral norm version of this Column Subset Selection Problem. We also prove that [EQUATION] with the same probability. This Frobenius norm bound is only a factor of √k log k worse than the best previously existing existential result and is roughly O(√k!) better than the best previous algorithmic result (both of Deshpande et al. [11]) for the Frobenius norm version of this Column Subset Selection Problem.