On the theoretical link between LLL-reduction and Lambda-decorrelation

Abstract

The LLL algorithm, introduced by Lenstra et al. (Math Ann 261:515–534, 1982), plays a key role in many fields of applied mathematics. In particular, it is used as an effective numerical tool for preconditioning the integer least-squares problems arising in high-precision geodetic positioning and Global Navigation Satellite Systems (GNSS). In 1992, Teunissen developed a method for solving these nearest-lattice point (NLP) problems. This method is referred to as Lambda (for Least-squares AMBiguity Decorrelation Adjustment). The preconditioning stage of Lambda corresponds to its decorrelation algorithm. From an epistemological point of view, the latter was devised through an innovative statistical approach completely independent of the LLL algorithm. Recent papers pointed out some similarities between the LLL algorithm and the Lambda-decorrelation algorithm. We try to clarify this point in the paper. We first introduce a parameter measuring the orthogonality defect of the integer basis in which the NLP problem is solved, the LLL-reduced basis of the LLL algorithm, or the $\Lambda$-basis of the Lambda method. With regard to this problem, the potential qualities of these bases can then be compared. The $\Lambda$-basis is built by working at the level of the variance-covariance matrix of the float solution, while the LLL-reduced basis is built by working at the level of its inverse. As a general rule, the orthogonality defect of the $\Lambda$-basis is greater than that of the corresponding LLL-reduced basis; these bases are however very close to one another. To specify this tight relationship, we present a method that provides the dual LLL-reduced basis of a given $\Lambda$-basis. As a consequence of this basic link, all the recent developments made on the LLL algorithm can be applied to the Lambda-decorrelation algorithm. This point is illustrated in a concrete manner: we present a parallel $\Lambda$-type decorrelation algorithm derived from the parallel LLL algorithm of Luo and Qiao (Proceedings of the fourth international C$^*$ conference on computer science and software engineering. ACM Int Conf P Series. ACM Press, pp 93–101, 2012).