### Abstract

The Riemannian trust-region algorithm (RTR) is designed to optimize differentiable cost functions on Riemannian manifolds. It proceeds by iteratively optimizing local models of the cost function. When these models are exact up to second order, RTR boasts a quadratic convergence rate to critical points. In practice, building such models requires computing the Riemannian Hessian, which may be challenging. A simple idea to alleviate this difficulty is to approximate the Hessian using finite differences of the gradient. Unfortunately, this is a nonlinear approximation, which breaks the known convergence results for RTR. We propose RTR-FD: a modification of RTR which retains global convergence when the Hessian is approximated using finite differences. Importantly, RTR-FD reduces gracefully to RTR if a linear approximation is used. This algorithm is available in the Manopt toolbox.

Original language | English (US) |
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Title of host publication | Geometric Science of Information - 2nd International Conference, GSI 2015, Proceedings |

Editors | Frank Nielsen, Frank Nielsen, Frank Nielsen, Frederic Barbaresco, Frederic Barbaresco, Frank Nielsen |

Publisher | Springer Verlag |

Pages | 467-475 |

Number of pages | 9 |

ISBN (Print) | 9783319250397, 9783319250397 |

DOIs | |

State | Published - Jan 1 2015 |

Externally published | Yes |

Event | 2nd International Conference on Geometric Science of Information, GSI 2015 - Palaiseau, France Duration: Oct 28 2015 → Oct 30 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9389 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 2nd International Conference on Geometric Science of Information, GSI 2015 |
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Country | France |

City | Palaiseau |

Period | 10/28/15 → 10/30/15 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Keywords

- Convergence
- Manopt
- Optimization on manifolds
- RTR-FD

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## Cite this

*Geometric Science of Information - 2nd International Conference, GSI 2015, Proceedings*(pp. 467-475). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9389). Springer Verlag. https://doi.org/10.1007/978-3-319-25040-3_50