The algebraic eigenvalue problem
The algebraic eigenvalue problem
Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
Introduction to statistical pattern recognition (2nd ed.)
Introduction to statistical pattern recognition (2nd ed.)
Polynomial time approximation schemes for dense instances of NP-hard problems
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
MAX-CUT has a randomized approximation scheme in dense graphs
Random Structures & Algorithms
Latent semantic indexing: a probabilistic analysis
PODS '98 Proceedings of the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
A constant-factor approximation algorithm for the k-median problem (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Finding a large hidden clique in a random graph
proceedings of the eighth international conference on Random structures and algorithms
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
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
Random sampling and approximation of MAX-CSP problems
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
A randomized approximation scheme for metric MAX-CUT
Journal of Computer and System Sciences
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Pass efficient algorithms for approximating large matrices
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A Two-Round Variant of EM for Gaussian Mixtures
UAI '00 Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence
Approximation schemes for clustering problems
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
Learning Mixtures of Gaussians
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
A Sublinear Time Approximation Scheme for Clustering in Metric Spaces
FOCS '99 Proceedings of the 40th 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
Spectral Partitioning of Random Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Pattern Classification (2nd Edition)
Pattern Classification (2nd Edition)
Approximation schemes for Metric Bisection and partitioning
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
On clusterings: Good, bad and spectral
Journal of the ACM (JACM)
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Clustering Large Graphs via the Singular Value Decomposition
Machine Learning
A spectral algorithm for learning mixture models
Journal of Computer and System Sciences - Special issue on FOCS 2002
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Spectral norm of random matrices
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Tensor decomposition and approximation schemes for constraint satisfaction problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On Learning Mixtures of Heavy-Tailed Distributions
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Spectral Partitioning, Eigenvalue Bounds, and Circle Packings for Graphs of Bounded Genus
SIAM Journal on Computing
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
A divide-and-merge methodology for clustering
ACM Transactions on Database Systems (TODS)
Fast computation of low-rank matrix approximations
Journal of the ACM (JACM)
The geometry of logconcave functions and sampling algorithms
Random Structures & Algorithms
k-means++: the advantages of careful seeding
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Spectral clustering with limited independence
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Eigenvalues and graph bisection: An average-case analysis
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
The Spectral Method for General Mixture Models
SIAM Journal on Computing
Robust PCA and clustering in noisy mixtures
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
NP-hardness of Euclidean sum-of-squares clustering
Machine Learning
PAC learning axis-aligned mixtures of gaussians with no separation assumption
COLT'06 Proceedings of the 19th annual conference on Learning Theory
On spectral learning of mixtures of distributions
COLT'05 Proceedings of the 18th annual conference on Learning Theory
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
Spectral methods for matrices and tensors
Proceedings of the forty-second ACM symposium on Theory of computing
Effective principal component analysis
SISAP'12 Proceedings of the 5th international conference on Similarity Search and Applications
Clustering and outlier detection using isoperimetric number of trees
Pattern Recognition
A scalable approach to column-based low-rank matrix approximation
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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Spectral methods refer to the use of eigenvalues, eigenvectors, singular values, and singular vectors. They are widely used in Engineering, Applied Mathematics, and Statistics. More recently, spectral methods have found numerous applications in Computer Science to “discrete” as well as “continuous” problems. This monograph describes modern applications of spectral methods, and novel algorithms for estimating spectral parameters. In the first part of the monograph, we present applications of spectral methods to problems from a variety of topics including combinatorial optimization, learning, and clustering. The second part of the monograph is motivated by efficiency considerations. A feature of many modern applications is the massive amount of input data. While sophisticated algorithms for matrix computations have been developed over a century, a more recent development is algorithms based on “sampling on the fly” from massive matrices. Good estimates of singular values and low-rank approximations of the whole matrix can be provably derived from a sample. Our main emphasis in the second part of the monograph is to present these sampling methods with rigorous error bounds. We also present recent extensions of spectral methods from matrices to tensors and their applications to some combinatorial optimization problems.