Polynomial filtering in latent semantic indexing for information retrieval

Kokiopoulou, Effrosyni; Saad, Yousef

doi:10.1145/1008992.1009013

Cited by 29 publications

(27 citation statements)

References 10 publications

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“…The sequence satisfies a 3-term recurrence and the approximation can be directly expressed in this basis. This was the approach taken in [10,19].…”

Section: Filtered Conjugate Residual Polynomial Iterationsmentioning

confidence: 99%

“…This approach is a slight variation of the one presented in [10,19]. The main difference, is that the algorithms in [10,19] focus on the solution polynomial instead of the residual polynomial, i.e., they do not explicitly compute or exploit residual polynomials.…”

Section: Filtered Conjugate Residual Polynomial Iterationsmentioning

confidence: 99%

“…The main difference, is that the algorithms in [10,19] focus on the solution polynomial instead of the residual polynomial, i.e., they do not explicitly compute or exploit residual polynomials. However, the two algorithms are mathematically equivalent.…”

Section: Filtered Conjugate Residual Polynomial Iterationsmentioning

confidence: 99%

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Filtered Conjugate Residual‐type Algorithms with Applications

Saad¹

2006

SIAM J. Matrix Anal. & Appl.

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In a number of applications, certain computations to be done with a given matrix are performed by replacing this matrix by its best low rank approximation. This has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies. One such application is that of regularization where the righthand side of the original linear system is noisy or inaccurate while the coefficient matrix is very ill-conditioned. Solving such linear systems accurately is counter-productive as the noise tends to be amplified. A common remedy is to compute the pseudo-inverse solution in which the inverses of the smallest singular values are replaced by zeros or small quantities. A similar procedure is also used in methods related to Principal Component Analysis, such as in Latent Semantic Indexing in information retrieval. Here the low-rank approximation to the original matrix is used to analyze similarities with a given query vector. This paper presents a few conjugate-gradient like methods to provide solutions to these two types of problems by iterative procedures which utilize only matrix-vector products.

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“…The sequence satisfies a 3-term recurrence and the approximation can be directly expressed in this basis. This was the approach taken in [10,19].…”

Section: Filtered Conjugate Residual Polynomial Iterationsmentioning

confidence: 99%

Section: Filtered Conjugate Residual Polynomial Iterationsmentioning

confidence: 99%

See 1 more Smart Citation

Filtered Conjugate Residual‐type Algorithms with Applications

Saad¹

2006

SIAM J. Matrix Anal. & Appl.

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“…To bypass the truncated SVD computation, Kokiopoulou and Saad [22] introduced polynomial filtering techniques, and Erhel et al [14] and Saad [26] proposed algorithms for building good polynomials to use in such techniques. These methods all efficiently compute a sequence of vectors that progressively approximate the vector s k defined in (1.3), without resorting to the expensive SVD.…”

Section: Introductionmentioning

confidence: 99%

Divide and Conquer Strategies for Effective Information Retrieval

Chen

Saad²

2009

Proceedings of the 2009 SIAM International Conference on Data Mining

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The standard application of Latent Semantic Indexing (LSI), a well-known technique for information retrieval, requires the computation of a partial Singular Value Decomposition (SVD) of the term-document matrix. This computation becomes infeasible for large document collections, since it is very demanding both in terms of arithmetic operations and in memory requirements. This paper discusses two divide and conquer strategies, with the goal of alleviating these difficulties. Both strategies divide the document collection in subsets, perform relevance analysis on each subset, and conquer the analysis results to form the query response. Since each sub-problem resulting from the recursive division process has a smaller size, the processing of large scale document collections requires much fewer resources. In addition, the computation is highly parallel and can be easily adapted to a parallel computing environment. To reduce the computational cost, we perform the analysis on the subsets by using the Lanczos vectors instead of singular vectors as in the standard LSI method. This technique is far more efficient than the computation of the truncated SVD, while its accuracy is comparable. Experimental results confirm that the proposed divide and conquer strategies are effective for information retrieval problems.

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“…For example, the filter polynomial used here is borrowed from [28], and earlier variants were used in [12] and [6]. The use of the partial reorthogonalization Lanczos (PR-Lanczos) was suggested in [1].…”

mentioning

confidence: 99%

Computation of Large Invariant Subspaces Using Polynomial Filtered Lanczos Iterations with Applications in Density Functional Theory

Bekas¹,

Kokiopoulou²,

Saad³

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. The most expensive part of all electronic structure calculations based on density functional theory lies in the computation of an invariant subspace associated with some of the smallest eigenvalues of a discretized Hamiltonian operator. The dimension of this subspace typically depends on the total number of valence electrons in the system, and can easily reach hundreds or even thousands when large systems with many atoms are considered. At the same time, the discretization of Hamiltonians associated with large systems yields very large matrices, whether with planewave or real-space discretizations. The combination of these two factors results in one of the most significant bottlenecks in computational materials science. In this paper we show how to efficiently compute a large invariant subspace associated with the smallest eigenvalues of a symmetric/Hermitian matrix using polynomially filtered Lanczos iterations. The proposed method does not try to extract individual eigenvalues and eigenvectors. Instead, it constructs an orthogonal basis of the invariant subspace by combining two main ingredients. The first is a filtering technique to dampen the undesirable contribution of the largest eigenvalues at each matrix-vector product in the Lanczos algorithm. This technique employs a well-selected low pass filter polynomial, obtained via a conjugate residual-type algorithm in polynomial space. The second ingredient is the Lanczos algorithm with partial reorthogonalization. Experiments are reported to illustrate the efficiency of the proposed scheme compared to state-of-the-art implicitly restarted techniques.

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Polynomial filtering in latent semantic indexing for information retrieval

Cited by 29 publications

References 10 publications

Filtered Conjugate Residual‐type Algorithms with Applications

Filtered Conjugate Residual‐type Algorithms with Applications

Divide and Conquer Strategies for Effective Information Retrieval

Computation of Large Invariant Subspaces Using Polynomial Filtered Lanczos Iterations with Applications in Density Functional Theory

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