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