A Distance for HMMs based on Aggregated Wasserstein Metric and State Registration
This work addresses a domain-specific problem for researchers and practitioners in time series analysis and machine learning, offering a more accurate and efficient distance metric for GMM-HMMs, though it is incremental as it builds on existing optimal transport and Wasserstein metric concepts.
The authors tackled the problem of measuring dissimilarity between Hidden Markov Models with Gaussian state distributions by proposing the Aggregated Wasserstein distance, which uses optimal transport to match states and achieves improved accuracy and efficiency in time series retrieval and classification tasks compared to existing methods based on Kullback-Leibler divergence.
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.