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Fast Least Squares Approximation Using Tensor Products of Functions and Linear Forms

  • Alexander Zimmermann

Abstract

Least squares approximations with functions play an important role in many mathematical and computer scientific applications. When dealing with input data of multidimensional structure, the use of tensor products of functions for approximation is very common. This paper presents a new approach for the representation of general linear forms with the help of tensor products allowing to separate and sequentialize least squares problems with multidimensional data. This leads to an enormous saving in computation time for calculating the solution, since the amount of arithmetic operations for d-dimensional input data will decrease from about \(\left( {\frac{2}{3}{n^{3d}} + 2{n^{2d}}} \right)\)to\( \left( {\frac{2}{3}d{n^3} + 2d{n^{d + 1}}} \right)\) Furthermore, the vast practical relevance of this technique is very important. It lasts from interpolation over surface approximation to imaging function renction reconsteuction in arbitrary dimensions.

Keywords

Tensor Product Linear Form Multilinear Mapping Finite Dimensional Vector Space Penrose Inverse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [DP91]
    Dodson, Christoper T.J.; Poston, Timothy: Tensor Geometry. 2. Berlin: Springer, 1991(Graduate texts in mathematics)Google Scholar
  2. [LH74]
    Lawson, Charles L.; Hanson, Richard J.: Solving least squares problems. Englewood Cliffs, NJ: Prentice-Hall, 1974Google Scholar
  3. [RM71]
    Rao, C. R.; Mitra, Sujit K.: Generalized inverse of matrices and its applications. New York: Wiley, 1971 (Wiley series in probability and mathematical statistics)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Alexander Zimmermann
    • 1
  1. 1.University of PassauPassauGermany

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