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1
September 2009
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. Spectral Algorithms describes modern applications of spectral methods, ...
2
March 2009
Foundations and Trends® in Theoretical Computer Science: Volume 4 Issue 3–4, March 2009
Publisher: Now Publishers Inc.
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, ...
3
July 2008
SIAM Journal on Computing: Volume 38 Issue 3, June 2008
Publisher: Society for Industrial and Applied Mathematics
We present an algorithm for learning a mixture of distributions based on spectral projection. We prove a general property of spectral projection for arbitrary mixtures and show that the resulting algorithm is efficient when the components of the mixture are logconcave distributions in $\Re^n$ whose means are separated. The separation ...
Keywords:
singular value decomposition, principal component analysis, logconcave distributions, mixture models
4
January 2007
SODA '07: Proceedings of the eighteenth annual ACMSIAM symposium on Discrete algorithms
Publisher: Society for Industrial and Applied Mathematics
Bibliometrics:
Citation Count: 17
Downloads (6 Weeks): 1, Downloads (12 Months): 5, Downloads (Overall): 251
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We propose and investigate bimatrix games, whose (entrywise) sum of the payoff matrices of the two players is of rank k , where k is a constant. We will say the rank of such a game is k . For every fixed k , the class of rank k games ...
5
January 2007
SODA '07: Proceedings of the eighteenth annual ACMSIAM symposium on Discrete algorithms
Publisher: Society for Industrial and Applied Mathematics
Bibliometrics:
Citation Count: 2
Downloads (6 Weeks): 0, Downloads (12 Months): 5, Downloads (Overall): 347
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This paper considers the wellstudied problem of clustering a set of objects under a probabilistic model of data in which each object is represented as a vector over the set of features, and there are only k different types of objects. In general, earlier results (mixture models and "planted" problems ...
6
December 2006
ACM Transactions on Database Systems (TODS): Volume 31 Issue 4, December 2006
Publisher: ACM
Bibliometrics:
Citation Count: 28
Downloads (6 Weeks): 3, Downloads (12 Months): 25, Downloads (Overall): 1,487
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We present a divideandmerge methodology for clustering a set of objects that combines a topdown “divide” phase with a bottomup “merge” phase. In contrast, previous algorithms use either topdown or bottomup methods to construct a hierarchical clustering or produce a flat clustering using local search (e.g., k means). For the ...
Keywords:
Clustering, data mining, information retrieval
7
September 2006
ESA'06: Proceedings of the 14th conference on Annual European Symposium  Volume 14
Publisher: SpringerVerlag
In this paper, we analyze the second eigenvector technique of spectral partitioning on the planted partition random graph model, by constructing a recursive algorithm using the second eigenvectors in order to learn the planted partitions. The correctness of our algorithm is not based on the ratiocut interpretation of the second ...
8
July 2006
SIAM Journal on Computing: Volume 36 Issue 1, 2006
Publisher: Society for Industrial and Applied Mathematics
Motivated by applications in which the data may be formulated as a matrix, we consider algorithms for several common linear algebra problems. These algorithms make more efficient use of computational resources, such as the computation time, random access memory (RAM), and the number of passes over the data, than do ...
Keywords:
massive data sets, matrix multiplication, streaming models, Monte Carlo methods, randomized algorithms
9
July 2006
SIAM Journal on Computing: Volume 36 Issue 1, 2006
Publisher: Society for Industrial and Applied Mathematics
In many applications, the data consist of (or may be naturally formulated as) an $m \times n$ matrix $A$. It is often of interest to find a lowrank approximation to $A$, i.e., an approximation $D$ to the matrix $A$ of rank not greater than a specified rank $k$, where $k$ ...
Keywords:
randomized algorithms, massive data sets, Monte Carlo methods, singular value decomposition
10
July 2006
Combinatorics, Probability and Computing: Volume 15 Issue 4, July 2006
Publisher: Cambridge University Press
The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by ...
11
July 2006
SIAM Journal on Computing: Volume 36 Issue 1, 2006
Publisher: Society for Industrial and Applied Mathematics
In many applications, the data consist of (or may be naturally formulated as) an $m \times n$ matrix $A$ which may be stored on disk but which is too large to be read into random access memory (RAM) or to practically perform superlinear polynomial time computations on it. Two algorithms ...
Keywords:
Monte Carlo methods, CUR matrix decomposition, randomized algorithms, massive data sets
12
January 2006
SODA '06: Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm
Publisher: Society for Industrial and Applied Mathematics
Bibliometrics:
Citation Count: 9
Downloads (6 Weeks): 2, Downloads (12 Months): 5, Downloads (Overall): 274
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We present multiple pass streaming algorithms for a basic clustering problem for massive data sets. If our algorithm is allotted 2 l passes, it will produce an approximation with error at most ε using Õ ( k 3 /ε 2 / l ) bits of memory, the most critical resource ...
13
June 2005
COLT'05: Proceedings of the 18th annual conference on Learning Theory
Publisher: SpringerVerlag
We present an algorithm for learning a mixture of distributions based on spectral projection. We prove a general property of spectral projection for arbitrary mixtures and show that the resulting algorithm is efficient when the components of the mixture are logconcave distributions in $\Re^{n}$ whose means are separated. The separation ...
14
June 2005
PODS '05: Proceedings of the twentyfourth ACM SIGMODSIGACTSIGART symposium on Principles of database systems
Publisher: ACM
Bibliometrics:
Citation Count: 24
Downloads (6 Weeks): 0, Downloads (12 Months): 4, Downloads (Overall): 422
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We present a divideandmerge methodology for clustering a set of objects that combines a topdown "divide" phase with a bottomup "merge" phase. In contrast, previous algorithms either use topdown or bottomup methods to construct a hierarchical clustering or produce a flat clustering using local search (e.g., k means). Our divide ...
15
May 2005
STOC '05: Proceedings of the thirtyseventh annual ACM symposium on Theory of computing
Publisher: ACM
Bibliometrics:
Citation Count: 13
Downloads (6 Weeks): 1, Downloads (12 Months): 11, Downloads (Overall): 322
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The only general class of MAXrCSP problems for which Polynomial Time Approximation Schemes (PTAS) are known are the dense problems. In this paper, we give PTAS's for a much larger class of weighted MAXrCSP problems which includes as special cases the dense problems and, for r = 2, all metric ...
Keywords:
approximation scheme, tensor decomposition
16
November 2004
Journal of the ACM (JACM): Volume 51 Issue 6, November 2004
Publisher: ACM
Bibliometrics:
Citation Count: 95
Downloads (6 Weeks): 27, Downloads (12 Months): 141, Downloads (Overall): 2,205
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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, ...
Keywords:
sampling, Matrix algorithms, lowrank approximation
17
June 2004
Machine Learning: Volume 56 Issue 13
Publisher: Kluwer Academic Publishers
We consider the problem of partitioning a set of m points in the n dimensional Euclidean space into k clusters (usually m and n are variable, while k is fixed), so as to minimize the sum of squared distances between each point and its cluster center. This formulation is usually ...
Keywords:
kmeans clustering, randomized algorithms, Singular Value Decomposition
18
May 2004
Journal of the ACM (JACM): Volume 51 Issue 3, May 2004
Publisher: ACM
Bibliometrics:
Citation Count: 166
Downloads (6 Weeks): 11, Downloads (12 Months): 146, Downloads (Overall): 4,125
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We motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures. A simple recursive heuristic is shown to have polylogarithmic worstcase guarantees under the new measure. The main result of the article is the analysis of a popular spectral ...
Keywords:
Clustering, spectral methods, graph algorithms
19
September 2003
Journal of Computer and System Sciences  STOC 2002: Volume 67 Issue 2, September 2003
Publisher: Academic Press, Inc.
In a maximum r constraint satisfaction problem with variables { x 1 , x 2 , ... , x n }, we are given Boolean functions f 1 , f 2 , ..., f m each involving r of the n variables and are to find the maximum number of ...
20
January 2003
SODA '03: Proceedings of the fourteenth annual ACMSIAM symposium on Discrete algorithms
Publisher: Society for Industrial and Applied Mathematics
Bibliometrics:
Citation Count: 44
Downloads (6 Weeks): 2, Downloads (12 Months): 15, Downloads (Overall): 427
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