Efficient Modification of the Upper Triangular Square Root Matrix on Variable Reordering

In probabilistic state inference, we seek to estimate the state of an (autonomous) agent from noisy observations. It can be shown that, under certain assumptions, finding the estimate is equivalent to solving a linear least squares problem. Solving such a problem is done by calculating the upper tri...

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Veröffentlicht in:IEEE robotics and automation letters Jg. 6; H. 2; S. 675 - 682
Hauptverfasser: Elimelech, Khen, Indelman, Vadim
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Piscataway IEEE 01.04.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Autonomous agents
Factorization
Incremental least squares
Least squares method
Mathematical analysis
Matrices (mathematics)
Matrix decomposition
Optimization
parallel algorithms
Parallel processing
Permutations
Probabilistic inference
Probabilistic logic
Robots
Simultaneous localization and mapping
SLAM
Smoothing methods
sparse systems
Symmetric matrices
ISSN:2377-3766, 2377-3766
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Zusammenfassung:In probabilistic state inference, we seek to estimate the state of an (autonomous) agent from noisy observations. It can be shown that, under certain assumptions, finding the estimate is equivalent to solving a linear least squares problem. Solving such a problem is done by calculating the upper triangular matrix <inline-formula><tex-math notation="LaTeX">\boldsymbol R</tex-math></inline-formula> from the coefficient matrix <inline-formula><tex-math notation="LaTeX">\boldsymbol A</tex-math></inline-formula>, using the QR or Cholesky factorizations; this matrix is commonly referred to as the "square root matrix". In sequential estimation problems, we are often interested in periodic optimization of the state variable order, e.g., to reduce fill-in, or to apply a predictive variable ordering tactic; however, changing the variable order implies expensive re-factorization of the system. Thus, we address the problem of modifying an existing square root matrix <inline-formula><tex-math notation="LaTeX">\boldsymbol R</tex-math></inline-formula>, to convey reordering of the variables. To this end, we identify several conclusions regarding the effect of column permutation on the factorization, to allow efficient modification of <inline-formula><tex-math notation="LaTeX">\boldsymbol R</tex-math></inline-formula>, without accessing <inline-formula><tex-math notation="LaTeX">\boldsymbol A</tex-math></inline-formula> at all, or with minimal re-factorization. The proposed parallelizable algorithm achieves a significant improvement in performance over the state-of-the-art incremental Smoothing And Mapping (iSAM2) algorithm, which utilizes incremental factorization to update <inline-formula><tex-math notation="LaTeX">\boldsymbol R</tex-math></inline-formula>.
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ISSN:2377-3766
2377-3766
DOI:10.1109/LRA.2020.3048663