# 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

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**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