Using latent semantic analysis to improve access to textual information
CHI '88 Proceedings of the SIGCHI Conference on Human Factors in Computing Systems
The algorithmic aspects of the regularity lemma
Journal of Algorithms
Authoritative sources in a hyperlinked environment
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
Fast computation of low rank matrix approximations
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Pass efficient algorithms for approximating large matrices
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Sampling lower bounds via information theory
Proceedings of the thirty-fifth 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
The regularity lemma and approximation schemes for dense problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Fast Monte-Carlo Algorithms for Approximate Matrix Multiplication
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Clustering Large Graphs via the Singular Value Decomposition
Machine Learning
Tensor decomposition and approximation schemes for constraint satisfaction problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
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
How to get close to the median shape
Proceedings of the twenty-second annual symposium on Computational geometry
Fast computation of low-rank matrix approximations
Journal of the ACM (JACM)
How to get close to the median shape
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
Sampling from large matrices: An approach through geometric functional analysis
Journal of the ACM (JACM)
Efficient subspace approximation algorithms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Bound for the L2 Norm of Random Matrix and Succinct Matrix Approximation
ICCS '08 Proceedings of the 8th international conference on Computational Science, Part II
Towards theory of generic Principal Component Analysis
Journal of Multivariate Analysis
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
On selecting a maximum volume sub-matrix of a matrix and related problems
Theoretical Computer Science
Foundations and Trends® in Theoretical Computer Science
Spectral methods for matrices and tensors
Proceedings of the forty-second ACM symposium on Theory of computing
Generalized low-rank approximations of matrices revisited
IEEE Transactions on Neural Networks
Matrix completion from a few entries
IEEE Transactions on Information Theory
Matrix Completion from Noisy Entries
The Journal of Machine Learning Research
Coresets and sketches for high dimensional subspace approximation problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
A Randomized Algorithm for Principal Component Analysis
SIAM Journal on Matrix Analysis and Applications
Stochastic algorithms in linear algebra: beyond the Markov chains and von Neumann-Ulam scheme
NMA'10 Proceedings of the 7th international conference on Numerical methods and applications
Property testing
Property testing
Proceedings of the ACM SIGMETRICS joint international conference on Measurement and modeling of computer systems
ACM SIGMETRICS Performance Evaluation Review - Performance evaluation review
Efficient manifold ranking for image retrieval
Proceedings of the 34th international ACM SIGIR conference on Research and development in Information Retrieval
Multiplicative approximations of random walk transition probabilities
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Larger residuals, less work: active document scheduling for latent dirichlet allocation
ECML PKDD'11 Proceedings of the 2011 European conference on Machine learning and knowledge discovery in databases - Volume Part III
Using rich social media information for music recommendation via hypergraph model
ACM Transactions on Multimedia Computing, Communications, and Applications (TOMCCAP) - Special section on ACM multimedia 2010 best paper candidates, and issue on social media
More influence means less work: fast latent dirichlet allocation by influence scheduling
Proceedings of the 20th ACM international conference on Information and knowledge management
SIAM Journal on Scientific Computing
Optimal column-based low-rank matrix reconstruction
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
Column subset selection via sparse approximation of SVD
Theoretical Computer Science
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
A randomized solver for linear systems with exponential convergence
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
Exact matrix completion via convex optimization
Communications of the ACM
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
Beating randomized response on incoherent matrices
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Cluster indicator decomposition for efficient matrix factorization
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume Two
Drawing Large Graphs by Low-Rank Stress Majorization
Computer Graphics Forum
Active spectral clustering via iterative uncertainty reduction
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
Graph Sparsification by Effective Resistances
SIAM Journal on Computing
A Fast Algorithm for Fourier Continuation
SIAM Journal on Scientific Computing
A GPU-based approximate SVD algorithm
PPAM'11 Proceedings of the 9th international conference on Parallel Processing and Applied Mathematics - Volume Part I
Approximation error in regularized SVD-based Fourier continuations
Applied Numerical Mathematics
Low rank approximation and regression in input sparsity time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Journal of Scientific Computing
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 consider the problem of approximating a given m × n matrix A by another matrix of specified rank k, which is smaller than m and n. The Singular Value Decomposition (SVD) can be used to find the "best" such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this article, we develop an algorithm that is qualitatively faster, provided we may sample the entries of the matrix in accordance with a natural probability distribution. In many applications, such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix D* of rank at most k so that holds with probability at least 1 − δ (where |·|F is the Frobenius norm). The algorithm takes time polynomial in k,1/ε, log(1/δ) only and is independent of m and n. In particular, this implies that in constant time, it can be determined if a given matrix of arbitrary size has a good low-rank approximation.