As we have explained in §\[sec:related\], the co-occurrence probability matrix ${\boldsymbol{\varOmega}}$ admits a nonnegative matrix tri-factorization model . There are a number of additional equality constraints. Columns of ${\boldsymbol{M}}$ represent conditional distributions, so ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}$. Matrix ${\boldsymbol{\varTheta}}$ represents the joint distribution between two consecutive Markovian variables, therefore ${\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}=1$. Furthermore, we have that ${\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}$ and ${\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}$ represent $\Pr[X_t]$ and $\Pr[X_{t+1}]$ respectively, and since we assume that the Markov chain is stationary, they are the same, i.e., ${\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}={\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}$. Notice that this does not imply that ${\boldsymbol{\varTheta}}$ is symmetric, and in fact it is often not symmetric.

@huang2016nips considered a factorization model similar to in a different context, and showed that identifiability can be achieved under a reasonable assumption called *sufficiently scattered*, defined as follows.

\[def:suf\_scat\] Let $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})^*$ denote the polyhedral cone $\{{\boldsymbol{x}}:{\boldsymbol{M}}{\boldsymbol{x}}\geq 0\}$, and ${\mathcal{C}}$ denote the elliptical cone $\{{\boldsymbol{x}}:\|{\boldsymbol{x}}\|\leq{\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{x}}\}$. Matrix ${\boldsymbol{M}}$ is called **sufficiently scattered** if it satisfies that: (i) $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})^*\subseteq{\mathcal{C}}$, and (ii) $\operatorname{cone}({{\boldsymbol{M}}}^{{\!\top\!}})^\ast\cap{\rm bd}{\mathcal{C}}=\{\lambda {\bm e}_k:\lambda\geq 0,k=1,...,K\}$, where ${\rm bd}{\mathcal{C}}$ denotes the boundary of ${\mathcal{C}}$, $\{{\boldsymbol{x}}:\|{\boldsymbol{x}}\|={\boldsymbol{\mathit{1}}}^{{{\!\top\!}}}{\boldsymbol{x}}\}$.

The sufficiently scattered condition was first proposed in [@huang2014tsp] to establish uniqueness conditions for the widely used *nonnegative matrix factorization* (NMF). For the NMF model ${\boldsymbol{\varOmega}}=\bm{WH}^{{\!\top\!}}$, if both $\bm{W}$ and $\bm{H}$ are sufficiently scattered, then 