Matrix Methods in Data Mining and Pattern Recognition

Eldén, Lars

doi:10.1137/1.9780898718867

Cited by 266 publications

(195 citation statements)

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“…Higham [12] showed that existing spectral reordering algorithms can be much quicker and more effective. We note that very similar aims arise in many other application areas, including pattern recognition [22], data mining [7], high performance computing [30] and sparse matrix computations [6,15]. In this work, our aims are 1. to develop a spectral algorithm that reveals "regular lattice plus short cuts" in the case where the underlying regular lattice has a periodic, rather than linear, structure, 2. to devise a computational test that determines whether a network is inherently more linear or periodic.…”

Section: Definition 22mentioning

confidence: 88%

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Periodic reordering

Grindrod¹,

Higham²,

Kalna³

2009

IMA Journal of Numerical Analysis

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For many networks in nature, science and technology, it is possible to order the nodes so that most links are short-range, connecting nearneighbours, and relatively few long-range links, or shortcuts, are present. Given a network as a set of observed links (interactions), the task of finding an ordering of the nodes that reveals such a range dependent structure is closely related to some sparse matrix reordering problems arising in scientific computation. The spectral, or Fiedler vector, approach for sparse matrix reordering has successfully been applied to biological data sets, revealing useful structures and subpatterns. In this work we argue that a periodic analogue of the standard reordering task is also highly relevant. Here, rather than encouraging nonzeros only to lie close to the diagonal of a suitably ordered adjacency matrix, we also allow them to inhabit the off-diagonal corners. Indeed, for the classic small-world model of Watts and Strogatz (Nature, 1998) this type of periodic structure is inherent. We therefore devise and test a new spectral algorithm for periodic reordering. By generalizing the range-dependent random graph class of Grindrod (Phys. Rev. E, 2002) to the periodic case, we can also construct a computable likelihood ratio that suggests whether a given network is inherently linear or periodic. Tests on synthetic data show that the new algorithm can detect periodic structure, even in the presence of noise. Further experiments on real biological data sets then show that some networks are better regarded as periodic than linear. Hence, we find both qualitative (reordered networks plots) and quantitative (likelihood ratios) evidence of periodicity in biological networks. * All authors were supported by the EPSRC under grant number GR/S62383/01. † Department of Mathematics and Centre for Advanced Computing and Emerging Technologies, University of Reading, RG6 6AX, UK.‡ Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK. § Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK. BackgroundLarge, sparse networks arise naturally when we describe the interconnectedness of components in complex systems [1,19,28]. The need to extract useful information creates challenging computational problems that, at least in part, overlap with sparse linear algebra tasks dealt with by numerical analysts. In this work we look at a matrix reordering problem that arises naturally from recent work in network modelling and computational biology. The reordering comes with a twist-a periodic analogue of the more usual "envelope reduction" or "two-sum minimization" is required.The presentation is organized as follows. In the next section we outline some recent random graph models that motivate the inverse problem. In section 3 we give a brief overview of the use of spectral methods for graph reordering, based on the graph Laplacian. We then derive a spectral algorithm for the periodic reordering problem and illustrate its use on specially constructed test data. In sectio...

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Section: Definition 22mentioning

confidence: 88%

“…Spectral projection of the nodes into a low-dimensional space is itself a wellstudied problem, with many algorithmic variants [2,7,17,22,24,27,30]. Here we outline an approach based on the normalized Laplacian that we have found to be useful.…”

Section: Definition 22mentioning

confidence: 99%

Periodic reordering

Grindrod¹,

Higham²,

Kalna³

2009

IMA Journal of Numerical Analysis

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“…In [6], they describe that a good process of matching queries is when the intersection between the set of documents retrieved and the set of relevant documents is as large as possible and the number of irrelevant documents recovered is small. In this way, to measure the performance of an information retrieval system, one must evaluate the ability of the system to retrieve relevant information (recall) and to reduce irrelevant information (precision).…”

Section: Queries and Measures Of Performancementioning

confidence: 99%

“…This factorization exists for any matrix A, and numerical linear algebra texts commonly include it in their content [7,8]. Methods to calculate the SVD of dense and sparse matrices are well documented [1,6,7].…”

Section: Singular Value Decompositionmentioning

confidence: 99%

“…In this regard, according to Theorem 2, LSI via SVD uses the singular value decomposition to obtain a k-rank approximation of the original document term matrix A mÂn in order to eliminate the noise present in it and project the m terms, n documents, and queries in a k-dimensional space, where k ≪ min m; n ðÞ . It is important to keep in mind that document term matrices are commonly well conditional, that is, their singular values have no gaps and do not decay rapidly to zero, so a suitable k to truncate the SVD cannot be estimated [6]; experimentally, it has been concluded that for very large databases, k is taken between [100; 300] [3].…”

Section: Lsi Via Svdmentioning

confidence: 99%

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Cross - Language Information Retrieval Using Two Methods: LSI via SDD and LSI via SVD

Lopez¹,

Blanquicett²,

Medina³

2018

Multilingualism and Bilingualism

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This chapter presents a method for the recovery of bilingual information based on semidiscrete matrix decomposition (SDD); that is, the problem of retrieving information in two languages, Spanish and English, is studied when the queries are made only in Spanish. In it, four case studies that exhibit the performance of the use of the latent semantic index (LSI) via SDD method for cross-language information retrieval (CLIR) are displayed. Concurrently, these results are compared with those obtained by applying LSI via singular value decomposition (SVD). All experiments were performed from a bilingual database, built from the gospels of the Bible, which combines documents in Spanish and English. For this, a fusion strategy was used that increases the size of the database by 10%. It was found that in terms of errors, the methods are comparable, since equal results were obtained in 58.3% of the queries made. In addition, the methods presented a success rate of at least 65% in the task of retrieving relevant information in the two languages considered.

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2017

Applied Mathematics for the Analysis of Biomedical Data

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Matrix Methods in Data Mining and Pattern Recognition

Cited by 266 publications

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Periodic reordering

Periodic reordering

Cross - Language Information Retrieval Using Two Methods: LSI via SDD and LSI via SVD

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