# A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group

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## Abstract

**:**

## 1. Introduction

## 2. Geometry of the Lorentz Group

**Definition**

**1.**

**Lemma**

**1.**

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Proposition**

**1.**

**Proof of Proposition**

**1.**

**Remark**

**1.**

## 3. Optimization on the Lorentz Group

#### 3.1. Riemannian-Steepest-Descent Algorithm on the Lorentz Group

**Proposition**

**2.**

**Proof of Proposition**

**2.**

**Remark**

**2.**

**Remark**

**3.**

**Definition**

**2.**

**Theorem**

**2.**

**Proof of Theorem**

**2.**

#### 3.2. Extended Hamiltonian Algorithm on the Lorentz Group

## 4. Numerical Experiments

#### 4.1. Numerical Experiments on Averaging Two Lorentz Matrices

#### 4.2. Numerical Experiments on Averaging Several Lorentz Matrices

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Harris, W.F. Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5 × 5 augmented system matrix. Optom. Vis. Sci.
**1994**, 71, 282–285. [Google Scholar] [CrossRef] [PubMed] - Barachant, A.; Bonnet, S.; Congedo, M.; Jutten, C. Multi-class brain computer interface classification by Riemannian geometry. IEEE Trans. Bio. Med. Eng.
**2012**, 59, 920–928. [Google Scholar] [CrossRef] [PubMed][Green Version] - Moakher, M. Means and averaging in the group of rotations. SIAM J. Matrix Anal. A
**2002**, 24, 1–16. [Google Scholar] [CrossRef] - Mello, P.A. Averages on the unitary group and applications to the problem of disordered conductors. J. Phys. A
**1990**, 23, 4061–4080. [Google Scholar] [CrossRef] - Duan, X.M.; Sun, H.F.; Peng, L.Y. Riemannian means on special Euclidean group and unipotent matrices group. Sci. World J.
**2013**, 2013, 292787. [Google Scholar] [CrossRef] [PubMed] - Chakraborty, R.; Vemuri, B.C. Recursive Fréchet mean computation on the Grassmannian and its applications to computer vision. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), Santiago, Chile, 7–13 December 2015. [Google Scholar]
- Fiori, S.; Kaneko, T.; Tanaka, T. Tangent-bundle maps on the Grassmann manifold: Application to empirical arithmetic averaging. IEEE Trans. Signal Process.
**2015**, 63, 155–168. [Google Scholar] [CrossRef] - Kaneko, T.; Fiori, S.; Tanaka, T. Empirical arithmetic averaging over the compact Stiefel manifold. IEEE Trans. Signal Process.
**2013**, 61, 883–894. [Google Scholar] [CrossRef] - Pölitz, C.; Duivesteijn, W.; Morik, K. Interpretable domain adaptation via optimization over the Stiefel manifold. Mach. Learn.
**2016**, 104, 315–336. [Google Scholar] [CrossRef] - Fiori, S. Learning the Fréchet mean over the manifold of symmetric positive-definite matrices. Cogn. Comput.
**2009**, 1, 279–291. [Google Scholar] [CrossRef] - Moakher, M. A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. A
**2005**, 26, 735–747. [Google Scholar] [CrossRef] - Buchholz, S.; Sommer, G. On averaging in Clifford groups. In Computer Algebra and Geometric Algebra with Applications; Springer: Berlin/Heidelberg, Germany, 2005; Volume 3519, pp. 229–238. [Google Scholar]
- Kawaguchi, H. Evaluation of the Lorentz group Lie algebra map using the Baker-Cambell-Hausdorff formula. IEEE Trans. Magn.
**1999**, 35, 1490–1493. [Google Scholar] [CrossRef] - Heine, V. Group Theory in Quantum Mechanics; Dover: New York, NY, USA, 1993. [Google Scholar]
- Geyer, C.M. Catadioptric Projective Geometry: Theory and Applications. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, USA, 2002. [Google Scholar]
- Fiori, S. Extended Hamiltonian learning on Riemannian manifolds: Theoretical aspects. IEEE Trans. Neural Netw.
**2011**, 22, 687–700. [Google Scholar] [CrossRef] [PubMed] - Fiori, S. Extended Hamiltonian learning on Riemannian manifolds: Numerical aspects. IEEE Trans. Neural Netw. Learn. Syst.
**2012**, 23, 7–21. [Google Scholar] [CrossRef] [PubMed] - Zhang, X. Matrix Analysis and Application; Springer: Beijing, China, 2004. [Google Scholar]
- Andruchow, E.; Larotonda, G.; Recht, L.; Varela, A. The left invariant metric in the general linear group. J. Geom. Phys.
**2014**, 86, 241–257. [Google Scholar] [CrossRef] - Zacur, E.; Bossa, M.; Olmos, S. Multivariate tensor-based morphometry with a right-invariant Riemannian distance on GL
^{+}(n). J. Math. Imaging Vis.**2014**, 50, 19–31. [Google Scholar] [CrossRef] - Goldberg, K. The formal power series for log(e
^{x}e^{y}). Duke Math. J.**1956**, 23, 1–21. [Google Scholar] [CrossRef] - Newman, M.; So, W.; Thompson, R.C. Convergence domains for the Campbell-Baker-Hausdorff formula. Linear Algebra Appl.
**1989**, 24, 30–310. [Google Scholar] [CrossRef] - Thompson, R.C. Convergence proof for Goldberg’s expoential series. Linear Algebra Appl.
**1989**, 121, 3–7. [Google Scholar] [CrossRef] - Karcher, H. Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math.
**1977**, 30, 509–541. [Google Scholar] [CrossRef] - Gabay, D. Minimizing a differentiable function over a differentiable manifold. J. Optim. Theory App.
**1982**, 37, 177–219. [Google Scholar] [CrossRef] - Fiori, S. Solving minimal-distance problems over the manifold of real symplectic matrices. SIAM J. Matrix Anal. A
**2011**, 32, 938–968. [Google Scholar] [CrossRef] - Fiori, S. A Riemannian steepest descent approach over the inhomogeneous symplectic group: Application to the averaging of linear optical systems. Appl. Math. Comput.
**2016**, 283, 251–264. [Google Scholar] [CrossRef]

**Figure 1.**The criterion function $J\left(U\right)$ and the Frobenius norm of U during iteration, respectively.

**Figure 2.**Convergence comparison among the Riemannian-steepest-descent algorithm (RSDA), the extended Hamiltonian algorithm (EHA) and the Euclidean gradient algorithm (EGA) during iteration.

**Figure 3.**(

**a**) convergence comparison among the RSDA, the EHA and the EGA during iteration; (

**b**) the norm $\parallel {A}^{-1}{\nabla}_{A}f\parallel $ of the Riemannian gradient (17) during iteration by the Riemannian-steepest-descent algorithm (RSDA); (

**c**–

**d**) the discrepancy $D(A,{B}_{q})$ before iteration $(A={A}_{0})$ and after iteration $(A=\overline{A})$, respectively.

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Wang, J.; Sun, H.; Li, D. A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group. *Entropy* **2017**, *19*, 698.
https://doi.org/10.3390/e19120698

**AMA Style**

Wang J, Sun H, Li D. A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group. *Entropy*. 2017; 19(12):698.
https://doi.org/10.3390/e19120698

**Chicago/Turabian Style**

Wang, Jing, Huafei Sun, and Didong Li. 2017. "A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group" *Entropy* 19, no. 12: 698.
https://doi.org/10.3390/e19120698