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 position 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 states. The distance is defined meaningfully even for two HMMs that are estimated from data of different dimensionality, a situation that can arise due to missing variables. This distance quantifies the dissimilarity of GMM-HMMs by measuring both the difference between the two marginal GMMs and that between the two transition matrices. Our new distance is tested on tasks of retrieval, classification, and t-SNE visualization of time series. Experiments on both synthetic 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.

本论文提出了一个名为 Aggregated Wasserstein 的框架，用于计算两个具有高斯状态条件分布的隐马尔可夫模型之间的差异度量或距离，通过匹配 GMM-HMM 中的状态来实现对其进行注册，通过求解优化问题匹配分量，通过对两个 GMM 之间的 Wasserstein 距离的分量之间成本解决，该优化问题的解是两个 GMM 之间 Wasserstein 距离的快速近似解，该新的 Aggregated Wasserstein 距离是一种半度量，可以计算而不生成 Monte Carlo 样本、并且可以旋转，即不改变其含义，用于测量 GMM-HMM 之间的差异，同时还考虑了两个转移矩阵之间的差异，在时间序列检索、分类和 t-SNE 可视化任务中测试了新的距离，同时在与基于 Kullback-Leibler 散度的现有距离的比较中证明了准确性和效率上的优势。

隐马尔可夫模型的聚合Wasserstein指标和状态对齐注册