### Abstract

Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if the original manifold is not Euclidean. In this paper, we propose to learn the distance function directly on the manifold without embedding. We first provide a theoretical characterization of the distance function by its gradient field. Based on our theoretical analysis, we propose to first learn the gradient field of the distance function and then learn the distance function itself. Specifically, we set the gradient field of a local distance function as an initial vector field. Then we transport it to the whole manifold via heat flow on vector fields. Finally, the geodesic distance function can be obtained by requiring its gradient field to be close to the normalized vector field. Experimental results on both synthetic and real data demonstrate the effectiveness of our proposed algorithm.

Original language | English (US) |
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Title of host publication | 31st International Conference on Machine Learning, ICML 2014 |

Publisher | International Machine Learning Society (IMLS) |

Pages | 1346-1354 |

Number of pages | 9 |

ISBN (Electronic) | 9781634393973 |

State | Published - Jan 1 2014 |

Event | 31st International Conference on Machine Learning, ICML 2014 - Beijing, China Duration: Jun 21 2014 → Jun 26 2014 |

### Publication series

Name | 31st International Conference on Machine Learning, ICML 2014 |
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Volume | 2 |

### Other

Other | 31st International Conference on Machine Learning, ICML 2014 |
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Country | China |

City | Beijing |

Period | 6/21/14 → 6/26/14 |

### Fingerprint

### ASJC Scopus subject areas

- Artificial Intelligence
- Computer Networks and Communications
- Software

### Cite this

*31st International Conference on Machine Learning, ICML 2014*(pp. 1346-1354). (31st International Conference on Machine Learning, ICML 2014; Vol. 2). International Machine Learning Society (IMLS).