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Journal of the ACM (JACM)
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
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Competitive recommendation systems
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Spectral Partitioning of Random Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Tabulation based 4-universal hashing with applications to second moment estimation
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Finding frequent items in data streams
Theoretical Computer Science - Special issue on automata, languages and programming
Clustering Large Graphs via the Singular Value Decomposition
Machine Learning
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Matrix approximation and projective clustering via volume sampling
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Sampling algorithms for l2 regression and applications
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication
SIAM Journal on Computing
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)
Sampling from large matrices: An approach through geometric functional analysis
Journal of the ACM (JACM)
Subspace sampling and relative-error matrix approximation: column-row-based methods
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
The Spectral Method for General Mixture Models
SIAM Journal on Computing
Numerical linear algebra in the streaming model
Proceedings of the forty-first annual ACM symposium on Theory of computing
A fast and efficient algorithm for low-rank approximation of a matrix
Proceedings of the forty-first annual ACM symposium on Theory of computing
Sampling Algorithms and Coresets for $\ell_p$ Regression
SIAM Journal on Computing
A sparse Johnson: Lindenstrauss transform
Proceedings of the forty-second ACM symposium on Theory of computing
Fast Manhattan sketches in data streams
Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Faster least squares approximation
Numerische Mathematik
Fast moment estimation in data streams in optimal space
Proceedings of the forty-third annual ACM symposium on Theory of computing
Sparser Johnson-Lindenstrauss transforms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Low rank matrix-valued chernoff bounds and approximate matrix multiplication
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Approximating a gram matrix for improved kernel-based learning
COLT'05 Proceedings of the 18th annual conference on Learning Theory
On spectral learning of mixtures of distributions
COLT'05 Proceedings of the 18th annual conference on Learning Theory
A fast random sampling algorithm for sparsifying matrices
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
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
Fast matrix rank algorithms and applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Sketching via hashing: from heavy hitters to compressed sensing to sparse fourier transform
Proceedings of the 32nd symposium on Principles of database systems
Simple and deterministic matrix sketching
Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining
Sparsity lower bounds for dimensionality reducing maps
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Sparser Johnson-Lindenstrauss Transforms
Journal of the ACM (JACM)
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We design a new distribution over poly(r ε-1) x n matrices S so that for any fixed n x d matrix A of rank r, with probability at least 9/10, SAx2 = (1 pm ε)Ax2 simultaneously for all x ∈ Rd. Such a matrix S is called a subspace embedding. Furthermore, SA can be computed in O(nnz(A)) + ~O(r2ε-2) time, where nnz(A) is the number of non-zero entries of A. This improves over all previous subspace embeddings, which required at least Ω(nd log d) time to achieve this property. We call our matrices S sparse embedding matrices. Using our sparse embedding matrices, we obtain the fastest known algorithms for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and lp-regression: to output an x' for which Ax'-b2 ≤ (1+ε)minx Ax-b2 for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A)) + ~O(d3ε-2) time, and another in O(nnz(A)log(1/ε)) + ~O(d3log(1/ε)) time. (Here ~O(f) = f ⋅ logO(1)(f).) to obtain a decomposition of an n x n matrix A into a product of an n x k matrix L, a k x k diagonal matrix D, and a n x k matrix W, for which F{A - L D W} ≤ (1+ε)F{A-Ak}, where Ak is the best rank-k approximation, our algorithm runs in O(nnz(A)) + ~O(nk2 ε-4log n + k3ε-5log2n) time. to output an approximation to all leverage scores of an n x d input matrix A simultaneously, with constant relative error, our algorithms run in O(nnz(A) log n) + ~O(r3) time. to output an x' for which Ax'-bp ≤ (1+ε)minx Ax-bp for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A) log n) + poly(r ε-1) time, for any constant 1 ≤ p