Identifiability of Polytopic Matrix Factorization

Olivier Vu Thanh, Nicolas Gillis

Abstract: Polytopic matrix factorization (PMF) decomposes a given matrix as the product of two factors where the rows of the first factor belong to a given convex polytope and the columns of the second factor belong to another given convex polytope. In this paper we show that if the polytopes have certain invariant properties, and that if the rows of the first factor and the columns of the second factor are sufficiently scattered within their corresponding polytope, then this PMF is identifiable, that is, the factors are unique up to a signed permutation. The PMF framework is quite general, as it recovers other known structured matrix factorization models, and is highly customizable depending on the application. Hence, our result provides sufficient conditions that guarantee the identifiability of a large class of structured matrix factorization models.