Topics in matrix analysis
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Improved Approximation Algorithms for Large Matrices via Random Projections
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Deterministic Sparse Column Based Matrix Reconstruction via Greedy Approximation of SVD
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Efficient Volume Sampling for Row/Column Subset Selection
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Algebraic Complexity Theory
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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
A scalable approach to column-based low-rank matrix approximation
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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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We prove that for any real-valued matrix X ε Rmxn, and positive integers r ≥ k, there is a subset of r columns of X such that projecting X onto their span gives a [EQUATION]-approximation to best rank-k approximation of X in Frobenius norm. We show that the trade-off we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in O(rnmω log m) arithmetic operations where ω is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in O(rnm2) arithmetic operations.