represents ${\mathcal{C}}$, and the inner circle represents ${\mathcal{C}}^*$, again intersecting with the plane, respectively. The rows of ${\boldsymbol{M}}$ are scaled to sum up to one, and they are represented by black dots in Figure \[fig:geo-scattered\]. Their conical hull is represented by the shaded region. The polygon with dashed lines represents the dual of $\operatorname{cone}({\boldsymbol{M}}^{{\!\top\!}})$, which is indeed a subset of ${\mathcal{C}}$, and touches the boundary of ${\mathcal{C}}$ only at the coordinate vectors.

Figure \[fig:geo-separable\] shows a special case of sufficiently scattered called *separability*, which first appeared in [@donoho2004does] also to establish uniqueness of NMF. In this case, all the coordinate vectors appear in rows of ${\boldsymbol{M}}$, therefore $\operatorname{cone}({\boldsymbol{M}})$ equals the nonnegative orthant. It makes sense that this condition makes the identification problem easier, but it is also a very restrictive assumption. The sufficiently scattered condition, on the other hand, only requires that the shaded region contains the inner circle, as shown in Figure \[fig:geo-scattered\]. Intuitively this requires that the rows of ${\boldsymbol{M}}$ be “well scattered” in the probability simplex, but not to the extent of “separable”. Separability-based HMM identification has been considered in [@Barlier2015; @Glaude2015]. However, the way they construct second-order statistics is very different from ours. Figure \[fig:geo-no\_id\] shows 