the nonnegative decomposition is unique up to column permutation and scaling. Follow up work strengthened and extended the identifiability results based on this geometry inspired condition [@fu2015bss; @huang2016nips; @fu2017spl]. A similar tri-factorization model was considered in [@huang2016nips] in the context of bag-of-words topic modeling, and it was shown that among all feasible solutions of , if we find one that minimizes $|\det{\boldsymbol{\varTheta}}|$, then it recovers the ground-truth latent factors ${\boldsymbol{M}}$ and ${\boldsymbol{\varTheta}}$, assuming the ground-truth ${\boldsymbol{M}}$ is sufficiently scattered. In our present context, we therefore propose the following problem formulation:

\[prob:main\] $$\begin{aligned}
\operatorname*{\textrm{minimize}}_{{\boldsymbol{\varTheta}},{\boldsymbol{M}}}~~~ & |\det{\boldsymbol{\varTheta}}| \\
\textrm{subject to}~~~ & {\boldsymbol{\varOmega}}={\boldsymbol{M}}{\boldsymbol{\varTheta}}{\boldsymbol{M}}^{{\!\top\!}}, \label{eq:Omega}\\
& {\boldsymbol{\varTheta}}\geq0, {\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}={\boldsymbol{\varTheta}}^{{\!\top\!}}{\boldsymbol{\mathit{1}}}, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{\varTheta}}{\boldsymbol{\mathit{1}}}=1, \label{eq:Theta}\\
& {\boldsymbol{M}}\geq0, {\boldsymbol{\mathit{1}}}^{{\!\top\!}}{\boldsymbol{M}}={\boldsymbol{\mathit{1}}}^{{\!\top\!}}. \end{aligned}$$

Regarding Problem , we have the following identifiability result.

\[thm:unique\] [@huang2016nips] Suppose ${\boldsymbol{\varOmega}}$ is constructed as ${\boldsymbol{\varOmega}}={\boldsymbol{M}}_\natural{\boldsymbol{\varTheta}}_\natural{\boldsymbol{M}}_\natural^{{\!\top\!}}$, where ${\boldsymbol{M}}_\natural$ and ${\boldsymbol{\varTheta}}_\natural$ satisfy the constraints in , and in addition (i) $\operatorname{rank}({\boldsymbol{\varTheta}}_\natural)=K$ and (ii) ${\boldsymbol{M}}_\natural$ is . Let $({\boldsymbol{M}}_\star,{\boldsymbol{\varTheta}}_\star)$ be an optimal solution for , then there must exist a permutation matrix ${\boldsymbol{\varPi}}\in{\mathbf{R}}^{K\times K}$ such that $${\boldsymbol{M}}_\natural = {\boldsymbol{M}}_\star{\boldsymbol{\varPi}}, \qquad
{\boldsymbol{\varTheta}}_\natural = {\boldsymbol{\varPi}}^{{\!\top\!}}{\boldsymbol{\varTheta}}_\star{\boldsymbol{\varPi}}.$$

One may notice that in [@huang2016nips], there are no constraints on the core matrix ${\boldsymbol{\varTheta}}$ as we do in . In terms of identifiability, it is easy to 