According to the graphical model shown in Figure \[fig:hmm\], it is easy to see that given the value of $X_t$, $Y_t$ is conditionally independent of all the other variables, leading to the factorization $$\begin{aligned}
\label{eq:hmm_fac2}
\Pr[Y_t,Y_{t+1}] 
= \sum_{k,j=1}^{K}&\Pr[Y_t|X_t=x_k]\Pr[Y_{t+1}|X_{t+1}=x_j]\Pr[X_t=x_k,X_{t+1}=x_j]\end{aligned}$$ Let ${\boldsymbol{\varOmega}}\in{\mathbf{R}}^{N\times N}$, ${\boldsymbol{M}}\in{\mathbf{R}}^{N\times K}$, and ${\boldsymbol{\varTheta}}\in{\mathbf{R}}^{K\times K}$, with their elements defined as $$\begin{aligned}
\varOmega_{n\ell} &= \Pr[Y_t=y_n,Y_{t+1}=y_\ell],\\
M_{nk} &= \Pr[Y_t=y_n|X_t=x_k],\\
\varTheta_{kj} &= \Pr[X_t=x_k,X_{t+1}=x_j].\end{aligned}$$ Then, equations can be written compactly as $$\begin{aligned}
\label{eq:MTM}
{\boldsymbol{\varOmega}}= {\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{\!\top\!}}.\end{aligned}$$ Noticing that $\eqref{eq:MTM}$ is a nonnegative matrix tri-factorization with a number of inconsequential constraints for ${\boldsymbol{M}}$ and ${\boldsymbol{\varTheta}}$ to properly represent probabilities, @Vanluyten2008 [@Lakshminarayanan2010; @Cybenko2011] proposed using nonnegative matrix factorization (NMF) to estimate the HMM probabilities. However, NMF-based methods have a serious shortcoming in this context: the tri-factorization  is in general not unique, because it is fairly easy to find a nonsingular matrix ${\boldsymbol{Q}}$ such that both ${\boldsymbol{M}}{\boldsymbol{Q}}\geq0$ and ${\boldsymbol{Q}}^{-1}{\boldsymbol{\varTheta}}{\boldsymbol{Q}}^{-{{\!\top\!}}}\geq0$, and then $\widetilde{{\boldsymbol{M}}}={\boldsymbol{M}}{\boldsymbol{Q}}$ and $\widetilde{{\boldsymbol{\varTheta}}}={\boldsymbol{Q}}^{-1}{\boldsymbol{\varTheta}}{\boldsymbol{Q}}^{-{{\!\top\!}}}$ are equally good solutions in terms of reconstructing the co-occurrence matrix ${\boldsymbol{\varOmega}}$. When we use $({\boldsymbol{M}},{\boldsymbol{\varTheta}})$ and $(\widetilde{{\boldsymbol{M}}},\widetilde{{\boldsymbol{\varTheta}}})$ to perform HMM inference, such as estimating hidden states or predicting new emissions, the two models often yield completely different results, unless ${\boldsymbol{Q}}$ is a permutation matrix.

A number of works propose to use *tensor* 