A distance for HMMS based on aggregated wasserstein metric and state registration

Yukun Chen, Jianbo Ye, Jia Li

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

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 languageEnglish (US)
Title of host publicationComputer Vision - 14th European Conference, ECCV 2016, Proceedings
EditorsNicu Sebe, Bastian Leibe, Jiri Matas, Max Welling
PublisherSpringer Verlag
Pages451-466
Number of pages16
ISBN (Print)9783319464657
DOIs
StatePublished - Jan 1 2016

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9910 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Fingerprint

Wasserstein Metric
Gaussian distribution
Registration
Optimal Transport
Hidden Markov models
Time series
Wasserstein Distance
Dissimilarity Measure
Mixture Distribution
Kullback-Leibler Divergence
Gaussian Mixture
Gaussian Mixture Model
Transition Matrix
Dissimilarity
Synthetic Data
Marginal Distribution
Conditional Distribution
Markov Model
Costs
Permutation

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Chen, Y., Ye, J., & Li, J. (2016). A distance for HMMS based on aggregated wasserstein metric and state registration. In N. Sebe, B. Leibe, J. Matas, & M. Welling (Eds.), 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
Chen, Yukun ; Ye, Jianbo ; Li, Jia. / A distance for HMMS based on aggregated wasserstein metric and state registration. Computer Vision - 14th European Conference, ECCV 2016, Proceedings. editor / Nicu Sebe ; Bastian Leibe ; Jiri Matas ; Max Welling. Springer Verlag, 2016. pp. 451-466 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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Chen, Y, Ye, J & Li, J 2016, A distance for HMMS based on aggregated wasserstein metric and state registration. in N Sebe, B Leibe, J Matas & M Welling (eds), Computer Vision - 14th European Conference, ECCV 2016, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9910 LNCS, Springer Verlag, pp. 451-466. https://doi.org/10.1007/978-3-319-46466-4_27

A distance for HMMS based on aggregated wasserstein metric and state registration. / Chen, Yukun; Ye, Jianbo; Li, Jia.

Computer Vision - 14th European Conference, ECCV 2016, Proceedings. ed. / Nicu Sebe; Bastian Leibe; Jiri Matas; Max Welling. Springer Verlag, 2016. p. 451-466 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9910 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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N2 - 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.

AB - 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.

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M3 - Conference contribution

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Chen Y, Ye J, Li J. A distance for HMMS based on aggregated wasserstein metric and state registration. In Sebe N, Leibe B, Matas J, Welling M, editors, Computer Vision - 14th European Conference, ECCV 2016, Proceedings. Springer Verlag. 2016. p. 451-466. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-46466-4_27