  Polytopic matrix factorization (PMF) is a recently introduced matrix
decomposition method in which the data vectors are modeled as linear
transformations of samples from a polytope. The successful recovery of the
original factors in the generative PMF model is conditioned on the
"identifiability" of the chosen polytope. In this article, we investigate the
problem of determining the identifiability of a polytope. The identifiability
condition requires the polytope to be permutation-and/or-sign-only invariant.
We show how this problem can be efficiently solved by using a graph
automorphism algorithm. In particular, we show that checking only the
generating set of the linear automorphism group of a polytope, which
corresponds to the automorphism group of an edge-colored complete graph, is
sufficient. This property prevents checking all the elements of the permutation
group, which requires factorial algorithm complexity. We demonstrate the
feasibility of the proposed approach through some numerical experiments.
