### Abstract

We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time spot follows a Gaussian mixture distribution, a fact exploited to softly match, aka register, the states in two HMMs. We refer to such HMMs as Gaussian mixture model-HMM (GMM-HMM). The registration of states is inspired by the intrinsic relationship of optimal transport and the Wasserstein metric between distributions. Specifically, the components of the marginal GMMs are matched by solving an optimal transport problem where the cost between components is the Wasserstein metric for Gaussian distributions. The solution of the optimization problem is a fast approximation to the Wasserstein metric between two GMMs. The new Aggregated Wasserstein distance is a semi-metric and can be computed without generating Monte Carlo samples. It is invariant to relabeling or permutation of the states. This distance quantifies the dissimilarity of GMM-HMMs by measuring both the difference between the two marginal GMMs and the difference between the two transition matrices. Our new distance is tested on the tasks of retrieval and classification of time series. Experiments on both synthetic data and real data have demonstrated its advantages in terms of accuracy as well as efficiency in comparison with existing distances based on the Kullback-Leibler divergence.

Original language | English (US) |
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Title of host publication | Computer Vision - 14th European Conference, ECCV 2016, Proceedings |

Editors | Nicu Sebe, Bastian Leibe, Jiri Matas, Max Welling |

Publisher | Springer Verlag |

Pages | 451-466 |

Number of pages | 16 |

ISBN (Print) | 9783319464657 |

DOIs | |

State | Published - Jan 1 2016 |

### 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 | 9910 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Computer Vision - 14th European Conference, ECCV 2016, Proceedings*(pp. 451-466). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9910 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-46466-4_27