Vector alignment in matrix Lie groups
Abstract
Many measurements in physics determine a linear transformation from pairs of input and output vectors, where the transformation is known in advance to belong to a matrix Lie group, for example the Lorentz group for the four-momenta measured by two inertial observers, or the unitary group for the states produced by a quantum circuit. Among the classical groups, only the rotation and unitary groups have a well-known closed form solution, given by the Kabsch and Horn algorithms and by the polar decomposition. We extend our previously proposed Lie algebra method from the image of the matrix exponential to the entire Lie group. We derive the projection in closed form for the classical groups over the real and complex numbers, including the indefinite orthogonal and unitary, symplectic, spin, special Euclidean and similarity groups, and we give conditions under which the method is exact for noiseless data, with examples in which it fails otherwise and an extension by a data dependent involution which is exact on the whole group. For noisy data, we bound the error of the method to first order in the noise, and we show that, for the orthogonal and unitary groups, it agrees with the least squares solution over the group to first order when the input vectors are isotropic, but not otherwise. We apply the method to an eight-qubit quantum circuit and to the rigid registration of point clouds, and compare it with Gauss--Newton refinement and direct minimization. A Julia implementation is provided.
Related
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BibTeX
@misc{sha2026vector,
title = {{Vector alignment in matrix Lie groups}},
author = {Congzhou M Sha},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.30868},
eprint = {2606.30868},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30868},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Vector alignment in matrix Lie groups
AU - Congzhou M Sha
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2606.30868
UR - https://arxiv.org/abs/2606.30868
AB - Many measurements in physics determine a linear transformation from pairs of input and output vectors, where the transformation is known in advance to belong to a matrix Lie group, for example the Lorentz group for the four-momenta measured by two inertial observers, or the unitary group for the states produced by a quantum circuit. Among the classical groups, only the rotation and unitary groups have a well-known closed form solution, given by the Kabsch and Horn algorithms and by the polar decomposition. We extend our previously proposed Lie algebra method from the image of the matrix exponential to the entire Lie group. We derive the projection in closed form for the classical groups over the real and complex numbers, including the indefinite orthogonal and unitary, symplectic, spin, special Euclidean and similarity groups, and we give conditions under which the method is exact for noiseless data, with examples in which it fails otherwise and an extension by a data dependent involution which is exact on the whole group. For noisy data, we bound the error of the method to first order in the noise, and we show that, for the orthogonal and unitary groups, it agrees with the least squares solution over the group to first order when the input vectors are isotropic, but not otherwise. We apply the method to an eight-qubit quantum circuit and to the rigid registration of point clouds, and compare it with Gauss--Newton refinement and direct minimization. A Julia implementation is provided.
ER -