Some applications of the rank revealing QR factorization
SIAM Journal on Scientific and Statistical Computing
Estimating the largest eigenvalues by the power and Lanczos algorithms with a random start
SIAM Journal on Matrix Analysis and Applications
Efficient algorithms for computing a strong rank-revealing QR factorization
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
EM algorithms for PCA and SPCA
NIPS '97 Proceedings of the 1997 conference on Advances in neural information processing systems 10
Latent semantic indexing: a probabilistic analysis
Journal of Computer and System Sciences - Special issue on the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on principles of database systems
Fast computation of low rank matrix approximations
STOC '01 Proceedings of the thirty-third 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
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Condition Numbers of Gaussian Random Matrices
SIAM Journal on Matrix Analysis and Applications
Matrix approximation and projective clustering via volume sampling
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix
SIAM Journal on Computing
SIAM Journal on Computing
Improved Approximation Algorithms for Large Matrices via Random Projections
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Fast computation of low-rank matrix approximations
Journal of the ACM (JACM)
Less is More: Sparse Graph Mining with Compact Matrix Decomposition
Statistical Analysis and Data Mining
Relative-Error $CUR$ Matrix Decompositions
SIAM Journal on Matrix Analysis and Applications
Numerical linear algebra in the streaming model
Proceedings of the forty-first annual ACM symposium on Theory of computing
An experimental evaluation of a Monte-Carlo algorithm for singular value decomposition
PCI'01 Proceedings of the 8th Panhellenic conference on Informatics
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
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Principal component analysis (PCA) requires the computation of a low-rank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rank-deficient approximation is at most a few digits (measured in the spectral norm, relative to the spectral norm of the matrix being approximated). In such circumstances, efficient algorithms have not come with guarantees of good accuracy, unless one or both dimensions of the matrix being approximated are small. We describe an efficient algorithm for the low-rank approximation of matrices that produces accuracy that is very close to the best possible accuracy, for matrices of arbitrary sizes. We illustrate our theoretical results via several numerical examples.