Abstract
In this paper, we present a new ridge estimation method for solving rank-deficient least squares problems, in which a rank-deficient matrix is regarded as an almost rank-deficient. First, we give an algebraic derivation that the optimal solution can in fact be obtained by solving a related regularized problem on the optimal worst-case residual. Second, we give a new iterative algorithm to solve ridge parameter and prove its convergence. Finally, examples are given to demonstrate the efficiency of our new method. It is shown that the proposed algorithm can not only assess the stability of solution but also use additional prior information to guarantee the uniqueness of solutions to the problem of rank-deficient free-network adjustment.
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Acknowledgements
This paper was supported by the National Natural Science Foundation of China (Project Nos. 41674009, 41574006, and 41674012). In Example 1 and Example 2, the observation data of side length and angle are obtained by their true values add random errors, in which the true values are obtained directly from the true coordinates (see Tables 1 and 7).
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Wenna LI and Caihua DENG, Yingchun SONG and Xianqiang CUI. The first draft of the manuscript was written by Yingchun SONG and Xianqiang CUI commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Song, Y., Li, W., Deng, C. et al. A new ridge estimation method on rank-deficient adjustment model. Acta Geod Geophys 57, 1–22 (2022). https://doi.org/10.1007/s40328-021-00366-0
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DOI: https://doi.org/10.1007/s40328-021-00366-0