Synthetic Data
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In this section we validate the identifiability performance on synthetic data. In this case, the underlying transition and emission probabilities are generated synthetically, and we compare them with the estimated ones to evaluate performance. The simulations are conducted in MATLAB using the HMM toolbox, which includes functions to generate observation sequences given transition and emission probabilities, as well as an implementation of the Baum-Welch algorithm [@baum1970maximization], i.e., the EM algorithm, to estimate the transition and emission probabilities using the observations. Unfortunately, even for some moderate problem sizes we considered, the streamlined MATLAB implementation of the Baum-Welch algorithm was not able to execute within reasonable amount of time, so its performance is not included here. For the baselines, we compare with the plain NMF approach using multiplicative update [@Vanluyten2008] and the tensor CPD approach [@Sharan2017] using simultaneous diagonalization with Tensorlab [@tensorlab3.0]. Since we work with empirical distributions instead of exact probabilities, the result of the simultaneous diagonalization is not going to be optimal. We therefore use it to initialize the EM algorithm for fitting a nonnegative tensor factorization with KL divergence loss [@shashanka2008probabilistic] for refinement.

We focus on the cases when the number of hidden states $K$ is 