Underdetermined Polytopic Matrix Factorization: Identifiability via Determinant Maximization and Face-Span Geometry

Structured matrix factorization supports analytical representation learning by recovering latent vectors from unknown linear mixtures through structural constraints. Nonnegativity, sparsity, and antisparsity motivate nonnegative matrix factorization, sparse component analysis, and bounded component analysis, respectively. Polytopic matrix factorization (PMF) unifies these approaches through a prescribed polytopic domain, allowing different combinations of structural properties, e.g., nonnegative sparse coefficients with signed, amplitude bounded coefficients in the same representation. Existing PMF identifiability, however, requires at least as many measurements as latent components. We extend PMF to the underdetermined regime, with fewer measurements than latent components, while retaining its source correlation determinant-maximization (Det-Max) criterion. Our approach combines PMF’s maximum-volume inscribed ellipsoid based sufficient scattering condition with a face-span skeleton formed by intersecting the polytope with linear spans of low-dimensional faces. Under explicit assumptions on cell population, relations among face spans, and local injectivity of the mixing matrix, we establish finite-sample exact recovery from noiseless observations: every global maximizer recovers the mixing matrix and all latent samples up to linear automorphism of the polytope. We derive sufficient measurement dimension conditions determined by the face-span geometry, demonstrate that Det-Max can exclude coordinate-mixing ambiguities, and establish a vanishing-regularization characterization of Det-Max through a source-entropy and reconstruction objective. These results provide a geometric foundation for underdetermined PMF, with implications for overcomplete dictionary learning with structural constraints tailored to different groups of latent coefficients.