### Abstract

Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.

Original language | English |
---|---|

Article number | 292787 |

Journal | The Scientific World Journal |

Volume | 2013 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Biochemistry, Genetics and Molecular Biology(all)
- Environmental Science(all)
- Medicine(all)

### Cite this

*The Scientific World Journal*,

*2013*, [292787]. https://doi.org/10.1155/2013/292787

**Riemannian means on special euclidean group and unipotent matrices group.** / Duan, Xiaomin; Sun, Huafei; Peng, Linyu.

Research output: Contribution to journal › Article

*The Scientific World Journal*, vol. 2013, 292787. https://doi.org/10.1155/2013/292787

}

TY - JOUR

T1 - Riemannian means on special euclidean group and unipotent matrices group

AU - Duan, Xiaomin

AU - Sun, Huafei

AU - Peng, Linyu

PY - 2013

Y1 - 2013

N2 - Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.

AB - Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.

UR - http://www.scopus.com/inward/record.url?scp=84887446672&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887446672&partnerID=8YFLogxK

U2 - 10.1155/2013/292787

DO - 10.1155/2013/292787

M3 - Article

C2 - 24282378

AN - SCOPUS:84887446672

VL - 2013

JO - The Scientific World Journal

JF - The Scientific World Journal

SN - 2356-6140

M1 - 292787

ER -