On the exact and efficient solution of the Huber function for measurement applications

Abstract

The findings of this paper are summarized as follows: (1) We point out that the recently published conclusion that Iteratively Reweighted Least-Squares (IRLS) cannot guarantee the global solution of the well-known Huber function is incorrect. We present that the Huber function has only one minimum and IRLS will obtain the exact minimum unless it is incorrectly implemented. (2) We revisit the variants of the IRLS algorithm and compare them with the exact IRLS algorithm in the geodetic free trilateration networks adjustment. Notably, the pointwise differences between the variants of IRLS and the exact IRLS are even beyond the differences between the Least-Squares solution and the exact IRLS for over 75% points. (3) For rigorously and efficiently solving the Huber M−estimation problem, we propose the Quasi-Newton approach and its limited memory version. The time expense of our proposed algorithm is quite low compared with the time expense of the existing exact IRLS in the cases of a large number of unknown parameters and observations. Our results show that the computation efficiency is improved by one or two orders of magnitude.