  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.
