Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Optimization of the Values of the Right-Hand Sides of Boundary Conditions with Point and Integral Terms for the ODE System Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Nonlinear Least Squares Solver for Evaluating Canonical Tensor Decomposition

Author : Igor Kaporin

Published in: Optimization and Applications

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Nonlinear least squares iterative solver developed earlier by the author is applied for numerical solution of special system of multilinear equations arising in the problem of canonical tensor decomposition. The proposed algorithm is based on easily parallelizable computational kernels such as matrix-vector multiplications and elementary vector operations and therefore has a potential for a quite efficient implementation on modern high-performance computers. The results of numerical testing presented for certain examples of large scale dense and medium size sparse 3D tensors found in existing literature seem very competitive with respect to computational costs involved.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Optimal Combination of Tensor Optimization Methods
next chapter PCGLNS: A Heuristic Solver for the Precedence Constrained Generalized Traveling Salesman Problem
Literature
1.
Acar, E., Dunlavy, D.M., Kolda, T.G.: A scalable optimization approach for fitting canonical tensor decompositions. J. Chemometr. 25(2), 67–86 (2011)CrossRef
2.
Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Math. 16(1), 1–3 (1966)MathSciNetCrossRef
3.
Bellavia, S., Gratton, S., Riccietti, E.: A Levenberg-Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients. Numer. Math. 140(3), 791–825 (2018)MathSciNetCrossRef
4.
Bellavia, S., Morini, B.: Strong local convergence properties of adaptive regularized methods for nonlinear least squares. IMA J. Numer. Anal. 35(2), 947–968 (2015)MathSciNetCrossRef
5.
Brent, R.P.: Algorithms for matrix multiplication (No. STAN-CS-70-157). Stanford University CA Department of Computer Science (1970)
6.
Kaporin, I.E.: Esimating global convergence of inexact Newton methods via limiting stepsize along normalized direction. Rep. 9329, July 1993, Department of Mathematics, Catholic University of Nijmegen, Nijmegen, The Netherlands, 8p. (1993)
7.
Kaporin, I.E.: The use of preconditioned Krylov subspaces in conjugate gradient type methods for the solution of nonlinear least square problems. (Russian) Vestnik Mosk. Univ., Ser. 15 (Computational Math. and Cybernetics) N 3, 26–31 (1995)
8.
Kaporin, I.E., Axelsson, O.: On a class of nonlinear equation solvers based on the residual norm reduction over a sequence of affine subspaces. SIAM J. Sci. Comput. 16(1), 228–249 (1994)MathSciNetCrossRef
9.
Kaporin, I.: Using Chebyshev polynomials and approximate inverse triangular factorizations for preconditioning the conjugate gradient method. Comput. Math. Math. Phys. 52(2), 169–193 (2012)MathSciNetCrossRef
10.
Kaporin, I.: Preconditioned subspace descent methods for the solution of nonlinear systems of equations. In: Jacimovic, M., et al. (eds.) Proceedings of X International Conference on Optimization and Applications (OPTIMA 2019). Communications in Computer and Information Science (CCIS). Springer Nature Switzerland AG (2019)
11.
Kaporin, I.: Preconditioned subspace descent method for nonlinear systems of equations. Open Comput. Sci. 10(1), 71–81 (2020)CrossRef
12.
13.
Kazeev, V.A., Tyrtyshnikov, E.E.: Structure of the Hessian matrix and an economical implementation of Newton’s method in the problem of canonical approximation of tensors. Comput. Math. Math. Phys. 50(6), 927–945 (2010)MathSciNetCrossRef
14.
Oseledets, I.V., Savostyanov, D.V.: Minimization methods for approximating tensors and their comparison. Comput. Math. Math. Phys. 46(10), 1641–1650 (2006)MathSciNetCrossRef
15.
Sterck, H.D., Winlaw, M.: A nonlinearly preconditioned conjugate gradient algorithm for rank-R canonical tensor approximation. Numer. Linear Algebra Appl. 22(3), 410–432 (2015)MathSciNetCrossRef
16.
Sterck, H.D., Miller, K.: An adaptive algebraic multigrid algorithm for low-rank canonical tensor decomposition. SIAM J. Sci. Comput. 35(1), B1–B24 (2013)MathSciNetCrossRef
17.
Yuan, Y.X.: Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numer. Algebra Control Optim. 1(1), 15–34 (2011)MathSciNetCrossRef
18.
Yu, L., Barbot, J.P., Zheng, G., Sun, H.: Compressive sensing with chaotic sequence. IEEE Signal Process. Lett. 17(8), 731–734 (2010)CrossRef
Metadata
Title
Nonlinear Least Squares Solver for Evaluating Canonical Tensor Decomposition
Author
Igor Kaporin
Copyright Year
2020
Book
Optimization and Applications

Print ISBN: 978-3-030-62866-6

Electronic ISBN: 978-3-030-62867-3

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-62867-3_14

Premium Partner

    Image Credits