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Axioms
Special Issues
Numerical Linear Algebra with Applications in Data Analysis
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Numerical Linear Algebra with Applications in Data Analysis

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: closed (31 January 2022) | Viewed by 3714
Please contact the Guest Editor or the Journal Editor for any queries about the scope, discount, submission procedure and publication process.

Special Issue Editor

Prof. Dr. Doulaye Dembélé

E-Mail Website
Guest Editor
Microarray Platform, IGBMC, CNRS-INSERM-UdS, 1 rue Laurent Fries, Parc d'Innovation, 67400 Illkirch, France
Interests: data analysis; signal processing; pattern recognition; advanced machine learning; statistical analysis; machine learning

Special Issue Information

Dear Colleagues,

Linear algebra deals with vector and matrices, which allows us to compactly manipulate data objects. On the one hand, vector and matrices are used in many mathematical theories: probability, statistics, optimization, learning, etc.; on the other hand, numerical linear algebra is used in many applications: solving linear systems of equations, linear regression in data classification, principal component analysis in data analysis, inversion of the Hessian matrix in the Newton–Raphson optimization method, ranking individuals using their social network interaction graph data, etc. Innovative numerical linear algebra tools are still needed to cope with the large amounts of data encountered in today’s life applications.

The purpose of this Special Issue is to gather a collection of articles reflecting new trends in numerical linear algebra with applications in data analysis. Some topics of interest are: vector space, matrix decomposition or factorization, accuracy, efficient computation, modeling, numerical stability and convergence. We welcome original research papers and review articles related to numerical linear algebra in the broad sense.

Prof. Dr. Doulaye Dembélé
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • vector spaces, range and subspace
  • vector and matrix norms
  • linear system of equations
  • least squares and total least squares
  • eigenvalues and singular values
  • matrix decomposition methods
  • positive and nonnegative matrices
  • stochastic matrices
  • special matrices and applications
  • large-scale systems and sparse matrices
  • direct and iterative methods

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Published Papers (1 paper)

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Research

18 pages, 304 KiB  
Article
The Smallest Singular Value Anomaly and the Condition Number Anomaly
by Achiya Dax
Axioms 2022, 11(3), 99; https://doi.org/10.3390/axioms11030099 - 25 Feb 2022
Cited by 3 | Viewed by 2876
Abstract
Let A be an arbitrary matrix in which the number of rows, m, is considerably larger than the number of columns, n. Let the submatrix Ai,i=1,…,m, be composed of the first [...] Read more.
Let A be an arbitrary matrix in which the number of rows, m, is considerably larger than the number of columns, n. Let the submatrix Ai,i=1,…,m, be composed of the first i rows of A. Let βi denote the smallest singular value of Ai, and let ki denote the condition number of Ai. In this paper, we examine the behavior of the sequences β1,…,βm, and k1,…,km. The behavior of the smallest singular values sequence is somewhat surprising. The first part of this sequence, β1,…,βn, is descending, while the second part, βn,…,βm, is ascending. This phenomenon is called “the smallest singular value anomaly”. The sequence of the condition numbers has a different character. The first part of this sequence, k1,…,kn, always ascends toward kn, which can be very large. The condition number anomaly occurs when the second part, kn,…,km, descends toward a value of km, which is considerably smaller than kn. It is shown that this is likely to happen whenever the rows of A satisfy two conditions: all the rows are about the same size, and the directions of the rows scatter in some random way. These conditions hold in a wide range of random matrices, as well as other types of matrices. The practical importance of these phenomena lies in the use of iterative methods for solving large linear systems, since several iterative solvers have the property that a large condition number results in a slow rate of convergence, while a small condition number yields fast convergence. Consequently, a condition number anomaly leads to a similar anomaly in the number of iterations. The paper ends with numerical experiments that illustrate the above phenomena. Full article
(This article belongs to the Special Issue Numerical Linear Algebra with Applications in Data Analysis)
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