Structure-preserving and rank-revealing QR-factorizations
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
Pass efficient algorithms for approximating large matrices
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Spectral Grouping Using the Nyström Method
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Algorithm 844: Computing sparse reduced-rank approximations to sparse matrices
ACM Transactions on Mathematical Software (TOMS)
SIAM Journal on Computing
On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning
The Journal of Machine Learning Research
Journal of Cognitive Neuroscience
Improved Nyström low-rank approximation and error analysis
Proceedings of the 25th international conference on Machine learning
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
Modeling wine preferences by data mining from physicochemical properties
Decision Support Systems
Clustered Nyström method for large scale manifold learning and dimension reduction
IEEE Transactions on Neural Networks
Efficient Volume Sampling for Row/Column Subset Selection
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Active learning using on-line algorithms
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
Near Optimal Column-Based Matrix Reconstruction
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Optimal column-based low-rank matrix reconstruction
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
Sampling methods for the Nyström method
The Journal of Machine Learning Research
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The CUR matrix decomposition and the Nyström approximation are two important low-rank matrix approximation techniques. The Nyström method approximates a symmetric positive semidefinite matrix in terms of a small number of its columns, while CUR approximates an arbitrary data matrix by a small number of its columns and rows. Thus, CUR decomposition can be regarded as an extension of the Nyström approximation. In this paper we establish a more general error bound for the adaptive column/row sampling algorithm, based on which we propose more accurate CUR and Nyström algorithms with expected relative-error bounds. The proposed CUR and Nyström algorithms also have low time complexity and can avoid maintaining the whole data matrix in RAM. In addition, we give theoretical analysis for the lower error bounds of the standard Nyström method and the ensemble Nyström method. The main theoretical results established in this paper are novel, and our analysis makes no special assumption on the data matrices.