2026/06/29 by Congzhou M Sha
Computer Science · Mathematics · #math.NA #cs.NA #hep-lat #math-ph #math.MP
Two observers of the same physical system may differ in gauge by a group element acting on their common vector representations, and recovering that element from finite, noisy paired observations is useful in both theory and experiment. The Kabsch and Horn algorithms solve this problem for rotated frames in \mathbb R3 (i.e. SO(3)); our earlier Lie algebra method extends it to the Lorentz group SO(3,1)+. Here we report explicit formulae for the Lie algebra method on the classical matrix Lie groups (GL(n), SL(n), SO(n), U(n), SO(p,q), Sp(n), Spin(n), and SE(n)) over the real and complex fields. The four steps (pseudoinverse, matrix logarithm, projection onto the Lie algebra, matrix exponential) are exact for noiseless data. Only the projection is group-dependent, and we show it yields the unique least squares-optimal element of the Lie algebra whenever its image lies in \mathfrak g and its residual is orthogonal to \mathfrak g. For noisy data the method is optimal only to leading order, so we add a quasi-Newton correction whose accuracy interpolates between the uncorrected method and direct least squares optimization. The projections, their optimality, and the identity underlying the correction are formally proven in Lean 4.31.0 with Mathlib, and numerical experiments are benchmarked in Julia.