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  • 1
    Digitale Medien
    Digitale Medien
    Accuracy of TSVD solutions computed from rank-revealing decompositions (1995)
    Springer
    Numerische Mathematik 70 (1995), S. 453-471 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Mathematics Subject Classification (1991):65F25, 65F30
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary. Rank-revealing decompositions are favorable alternatives to the singular value decomposition (SVD) because they are faster to compute and easier to update. Although they do not yield all the information that the SVD does, they yield enough information to solve various problems because they provide accurate bases for the relevant subspaces. In this paper we consider rank-revealing decompositions in computing estimates of the truncated SVD (TSVD) solution to an overdetermined system of linear equations $A x \approx b$ , where $A$ is numerically rank deficient. We derive analytical bounds which show how the accuracy of the solution is intimately connected to the quality of the subspaces.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050128
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  • 2
    Digitale Medien
    Digitale Medien
    Low-rank Orthogonal Decompositions for Information Retrieval Applications (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 301-327 
    Details
    ISSN: 1070-5325
    Schlagwort(e): information ; latent semantic indexing ; low-rank ; orthogonal ; matrices ; metrieval ; singular value decomposition ; sparse ; ULV and URV decompositions ; updating ; Engineering ; Engineering General
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: Current methods to index and retrieve documents from databases usually depend on a lexical match between query terms and keywords extracted from documents in a database. These methods can produce incomplete or irrelevant results due to the use of synonyms and polysemus words. The association of terms with documents (or implicit semantic structure) can be derived using large sparse {\it term-by-document} matrices. In fact, both terms and documents can be matched with user queries using representations in k-space (where 100 ≤ k ≤ 200) derived from k of the largest approximate singular vectors of these term-by-document matrices. This completely automated approach called latent semantic indexing or LSI, uses subspaces spanned by the approximate singular vectors to encode important associative relationships between terms and documents in k-space. Using LSI, two or more documents may be closeto each other in k-space (and hence meaning) yet share no common terms. The focus of this work is to demonstrate the computational advantages of exploiting low-rank orthogonal decompositions such as the ULV (or URV) as opposed to the truncated singular value decomposition (SVD) for the construction of initial and updated rank-k subspaces arising from LSI applications.
    Zusätzliches Material: 11 Ill.
    Materialart: Digitale Medien
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  • 3
    Digitale Medien
    Digitale Medien
    Orthogonal projection and total least squares (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 135-153 
    Details
    ISSN: 1070-5325
    Schlagwort(e): orthogonal projection ; numerical rank ; total least squares ; rank revealing QR factorization ; acute perturbation ; subspace angle ; Engineering ; Engineering General
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: Overdetermined linear systems often arise in applications such as signal processing and modern communication. When the overdetermined system of linear equations AX ≍ B has no solution, compatibility may be restored by an orthogonal projection method. The idea is to determine an orthogonal projection matrix P by some method M such that [à B̃] = P[A B], and ÃX = B̃ is compatible. Denote by XM the minimum norm solution to ÃX = B̃ using method M. In this paper conditions for compatibility of the lower rank approximation and subspace properties of à in relation to the nearest rank-k matrix to A are discussed. We find upper and lower bounds for the difference between the solution XM and the SVD-based total least squares (TLS) solution XSVD and also provide a perturbation result for the ordinary TLS method. These results suggest a new algorithm for computing a total least squares solution based on a rank revealing QR factorization and subspace refinement. Numerical simulations are included to illustrate the conclusions.
    Zusätzliches Material: 4 Tab.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/nla.1680020206
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  • 4
    Digitale Medien
    Digitale Medien
    Low-rank revealing UTV decompositions (1997)
    Springer
    Numerical algorithms 15 (1997), S. 37-55 
    Details
    ISSN: 1572-9265
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract A UTV decomposition of an m × n matrix is a product of an orthogonal matrix, a middle triangular matrix, and another orthogonal matrix. In this paper we present and analyze algorithms for computing updatable rank-revealing UTV decompositions that are efficient whenever the numerical rank of the matrix is much less than its dimensions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1019254318361
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  • 5
    Digitale Medien
    Digitale Medien
    UTV Tools: Matlab templates for rank-revealing UTV decompositions (1999)
    Springer
    Numerical algorithms 20 (1999), S. 165-194 
    Details
    ISSN: 1572-9265
    Schlagwort(e): rank-revealing decompositions ; rank deficiency ; numerical rank ; up- and downdating ; Matlab ; 65F25 ; 65F20
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract We describe a Matlab 5.2 package for computing and modifying certain rank-revealing decompositions that have found widespread use in signal processing and other applications. The package focuses on algorithms for URV and ULV decompositions, collectively known as UTV decompositions. We include algorithms for the ULLV decomposition, which generalizes the ULV decomposition to a pair of matrices. For completeness a few algorithms for computation of the RRQR decomposition are also included. The software in this package can be used as is, or can be considered as templates for specialized implementations on signal processors and similar dedicated hardware platforms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1019112103049
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