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Optimal alignment of Lorentz orientation and generalization to matrix Lie groups

Congzhou M Sha

TL;DR

This work tackles the problem of aligning 4-vector measurements across inertial frames by finding the optimal Lorentz transformation in $SO(3,1)_+$ that maps $\{v_{A,i}\}$ to $\{v_{B,i}\}$ under a Minkowski metric $\eta$. It first explains why Euclidean Procrustes methods (Kabsch, Horn) fail in this setting due to the indefinite inner product and noncompact group, and then introduces two viable solutions: (i) a direct nonlinear least-squares optimization over the Lorentz parameters (boosts and rotations) and (ii) a Lie-algebra projection method that solves an unconstrained problem and then projects back onto $SO(3,1)_+$ via the matrix logarithm and exponential, generalizable to other matrix Lie groups. Empirical results show both methods achieve comparable accuracy under ideal/noisy conditions, with the Lie-algebra approach offering major speed advantages and better scalability to other Lie groups. The work highlights practical implications for pose alignment in relativistic settings and as a general toolkit for manifold alignment in physics and geometry.

Abstract

There exist elegant methods of aligning point clouds in $\mathbb R^3$. Unfortunately, these methods fail to generalize to the case of Minkowski space, as we will show. Instead, we propose two solutions to the following problem: given inertial reference frames $A$ and $B$, and given (possibly noisy) measurements of a set of 4-vectors $\{v_i\}$ made in those reference frames with components $\{v_{A,i}\}$ and $\{v_{B,i}\}$, find the optimal Lorentz transformation $Λ$ such that $Λv_{A,i}=v_{B,i}$. The first method is direct least squares optimization through a parametrization of $SO(3,1)_+$ in terms of the familiar boost and rotation vectors. The second method takes a detour through the Lorentz algebra; in addition to being conceptually simple and possessing a computational advantage over the first method, it can easily be generalized to the alignment of vector representations in other matrix Lie groups.

Optimal alignment of Lorentz orientation and generalization to matrix Lie groups

TL;DR

This work tackles the problem of aligning 4-vector measurements across inertial frames by finding the optimal Lorentz transformation in that maps to under a Minkowski metric . It first explains why Euclidean Procrustes methods (Kabsch, Horn) fail in this setting due to the indefinite inner product and noncompact group, and then introduces two viable solutions: (i) a direct nonlinear least-squares optimization over the Lorentz parameters (boosts and rotations) and (ii) a Lie-algebra projection method that solves an unconstrained problem and then projects back onto via the matrix logarithm and exponential, generalizable to other matrix Lie groups. Empirical results show both methods achieve comparable accuracy under ideal/noisy conditions, with the Lie-algebra approach offering major speed advantages and better scalability to other Lie groups. The work highlights practical implications for pose alignment in relativistic settings and as a general toolkit for manifold alignment in physics and geometry.

Abstract

There exist elegant methods of aligning point clouds in . Unfortunately, these methods fail to generalize to the case of Minkowski space, as we will show. Instead, we propose two solutions to the following problem: given inertial reference frames and , and given (possibly noisy) measurements of a set of 4-vectors made in those reference frames with components and , find the optimal Lorentz transformation such that . The first method is direct least squares optimization through a parametrization of in terms of the familiar boost and rotation vectors. The second method takes a detour through the Lorentz algebra; in addition to being conceptually simple and possessing a computational advantage over the first method, it can easily be generalized to the alignment of vector representations in other matrix Lie groups.

Paper Structure

This paper contains 14 sections, 6 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Insights into pose alignment from the Kabsch and Horn algorithms
  3. The Kabsch and Horn algorithms
  4. Can we use the Kabsch or Horn algorithms as inspiration for Minkowski vector alignment?
  5. Proposed solutions to the Lorentz alignment problem
  6. Review of the Lorentz algebra and group
  7. Method 1: Direct nonlinear optimization
  8. Method 2: Projection onto the Lorentz group after unconstrained solving
  9. Results
  10. Ideal conditions with no noise
  11. Ideal conditions with no noise
  12. Varying the number of vectors given, noise, and averaging over arbitrary Lorentz transformations
  13. Discussion
  14. Quaternions

Key Result

Theorem 1

(Kabsch algorithm) Let there be two lists $\{\mathbf{v}_{A,i}\}$ and $\{\mathbf{v}_{B,i}\}$ of $n$ vectors in $N$ dimensions. Let $A$ and $B$ be $N\times n$ matrices where $A_{ji}=(\mathbf{v}_{A,i})_j$ and $B_{ji}=(\mathbf{v}_{B,i})_j$. Let the objective function be where $\|\cdot\|$ is the Frobenius norm. Given the singular value decomposition of $BA^T=U\Sigma V^T$. If $|UV^T|=1$, the rotation m

Figures (2)

  • Figure 1: The absolute Frobenius ($L^2$) norm of the estimated Lorentz matrix from frame $A$ to frame $B$ as compared to the actual Lorentz matrix ($\|\Lambda_{\text{est}}-\Lambda_{\text{GT}}\|_2$). The rows represent varying the noise $\epsilon$ present in the measurement of the vectors in frame $B$. The columns represent varying the number $n$ of vectors upon which the estimation was performed. Evidently, the Lie algebra method and the direct minimization method are equivalent in accuracy across a range of testing conditions, since their absolute error distributions largely overlap.
  • Figure 2: The absolute max ($L^\infty$) norm of the estimated Lorentz matrix from frame $A$ to frame $B$ as compared to the actual Lorentz matrix ($\|\Lambda_{\text{est}}-\Lambda_{\text{GT}}\|_text{max}$). The rows and columns represent the same variables as in Fig \ref{['fig:l2']}.

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • ...and 2 more