On lattice reduction algorithms for solving weighted integer least squares problems: comparative study

Decorrelation or reduction theory deals with identifying appropriate lattice bases that aid in accelerating integer search to find the optimal integer solution of the weighted integer least squares problem. Orthogonality defect has been widely used to measure the degree of orthogonality of the reduc...

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Veröffentlicht in:	GPS solutions Jg. 18; H. 1; S. 105 - 114
Hauptverfasser:	Jazaeri, Shahram, Amiri-Simkooei, Alireza, Sharifi, Mohammad Ali
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2014 Springer Nature B.V
Schlagworte:	Algorithms Ambiguity resolution (mathematics) Atmospheric Sciences Automotive Engineering Comparative studies Computer simulation Defects Earth and Environmental Science Earth Sciences Economic models Electrical Engineering Estimating techniques Geophysics/Geodesy Global positioning systems GPS Integers Least squares method Original Article Orthogonality Reduction Search process Space Exploration and Astronautics Space Sciences (including Extraterrestrial Physics Spatial data Upper bounds Seysen’s reduction algorithm LAMBDA decorrelation algorithm Integer least squares estimation Lattice theory Closest lattice point problem LLL decorrelation algorithm
ISSN:	1080-5370, 1521-1886
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Abstract	Decorrelation or reduction theory deals with identifying appropriate lattice bases that aid in accelerating integer search to find the optimal integer solution of the weighted integer least squares problem. Orthogonality defect has been widely used to measure the degree of orthogonality of the reduced lattice bases for many years. This contribution presents an upper bound for the number of integer candidates in the integer search process. This upper bound is shown to be a product of three factors: (1) the orthogonality defect, (2) the absolute value of the determinant of the inverse of the generator matrix of the lattice, and (3) the radius of the search space raised to the power of the dimension of the integer ambiguity vector. Four well-known decorrelation algorithms, namely LLL, LAMBDA, MLAMBDA, and Seysen, are compared. Many simulated data with varying condition numbers and dimensions as well as real GPS data show that the Seysen reduction algorithm reduces the condition number much better than the other algorithms. Also, the number of integer candidates, before and after the reduction process, is counted for all algorithms. Comparing the number of integer candidates, condition numbers, and orthogonality defect reveals that reducing the condition number and the orthogonality defect may not necessarily result in decreasing the number of integer candidates in the search process. Therefore, contrary to the common belief, reducing the orthogonality defect and condition number do not always result in faster integer least squares estimation. The results indicate that LAMBDA and MLAMBDA perform much better in reducing the number of integer candidates than the other two algorithms, despite having a larger orthogonality defect and condition number in some cases. Therefore, these two algorithms can speed up the integer least squares estimation problem in general and the integer ambiguity resolution problem in particular.
AbstractList	Decorrelation or reduction theory deals with identifying appropriate lattice bases that aid in accelerating integer search to find the optimal integer solution of the weighted integer least squares problem. Orthogonality defect has been widely used to measure the degree of orthogonality of the reduced lattice bases for many years. This contribution presents an upper bound for the number of integer candidates in the integer search process. This upper bound is shown to be a product of three factors: (1) the orthogonality defect, (2) the absolute value of the determinant of the inverse of the generator matrix of the lattice, and (3) the radius of the search space raised to the power of the dimension of the integer ambiguity vector. Four well-known decorrelation algorithms, namely LLL, LAMBDA, MLAMBDA, and Seysen, are compared. Many simulated data with varying condition numbers and dimensions as well as real GPS data show that the Seysen reduction algorithm reduces the condition number much better than the other algorithms. Also, the number of integer candidates, before and after the reduction process, is counted for all algorithms. Comparing the number of integer candidates, condition numbers, and orthogonality defect reveals that reducing the condition number and the orthogonality defect may not necessarily result in decreasing the number of integer candidates in the search process. Therefore, contrary to the common belief, reducing the orthogonality defect and condition number do not always result in faster integer least squares estimation. The results indicate that LAMBDA and MLAMBDA perform much better in reducing the number of integer candidates than the other two algorithms, despite having a larger orthogonality defect and condition number in some cases. Therefore, these two algorithms can speed up the integer least squares estimation problem in general and the integer ambiguity resolution problem in particular.
Author	Amiri-Simkooei, Alireza Sharifi, Mohammad Ali Jazaeri, Shahram
Author_xml	– sequence: 1 givenname: Shahram surname: Jazaeri fullname: Jazaeri, Shahram email: jazayeri@ut.ac.ir, jazaeri.s@nisoc.ir organization: Department of Surveying and Geomatics Engineering, College of Engineering, University of Tehran, Department of Surveying Engineering, National Iranian South Oil Company (NISOC) – sequence: 2 givenname: Alireza surname: Amiri-Simkooei fullname: Amiri-Simkooei, Alireza organization: Department of Surveying Engineering, Faculty of Engineering, University of Isfahan – sequence: 3 givenname: Mohammad Ali surname: Sharifi fullname: Sharifi, Mohammad Ali organization: Department of Surveying and Geomatics Engineering, College of Engineering, University of Tehran
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Cites_doi	10.1007/s00190-011-0490-y 10.1007/s00190-011-0501-z 10.1007/s10291-010-0192-6 10.1007/s00190-005-0004-x 10.1109/TIT.2002.800499 10.1007/BF00863419 10.1007/978-3-642-72011-6 10.1007/s001900100192 10.1007/BF01581144 10.1007/BF01202355 10.1007/s001900050269 10.1007/BF01442795 10.1007/BF01457454 10.1007/s001900050199 10.1109/MSP.2010.938758 10.1007/s10291-012-0276-6
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Keywords	Seysen’s reduction algorithm LAMBDA decorrelation algorithm Integer least squares estimation Lattice theory Closest lattice point problem LLL decorrelation algorithm
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SubjectTerms	Algorithms Ambiguity resolution (mathematics) Atmospheric Sciences Automotive Engineering Comparative studies Computer simulation Defects Earth and Environmental Science Earth Sciences Economic models Electrical Engineering Estimating techniques Geophysics/Geodesy Global positioning systems GPS Integers Least squares method Original Article Orthogonality Reduction Search process Space Exploration and Astronautics Space Sciences (including Extraterrestrial Physics Spatial data Upper bounds
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Title	On lattice reduction algorithms for solving weighted integer least squares problems: comparative study
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