Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability

We introduce Polytopic Matrix Factorization (PMF) as a novel data decomposition approach. In this new framework, we model input unknown linear transformations of some latent vectors drawn from polytope. sense, the article considers semi-structured model, in which matrix is modeled product full column rank and containing samples polytope its vectors. The choice reflects presumed features components their mutual relationships. As factorization criterion, propose determinant maximization (Det-Max) for sample autocorrelation sufficient condition identifiability, requires that convex hull contains maximum volume inscribed ellipsoid with particular tightness constraint. Based on Det-Max criterion proposed identifiability condition, show all polytopes satisfy symmetry restriction qualify PMF framework. Having infinitely many choices provides form flexibility characterizing particular, it possible to define heterogeneous features, enabling assignment attributes such nonnegativity sparsity at subvector level. offers examples illustrating connection between corresponding feature representations.