Generalized Polytopic Matrix Factorization

Abstract

Polytopic Matrix Factorization (PMF) is introduced as a flexible data decomposition tool with potential applications in unsupervised learning. PMF assumes a generative model where observations are lossless linear mixtures of some samples drawn from a particular polytope. Assuming that these samples are sufficiently scattered inside the polytope, a determinant maximization based criterion is used to obtain latent polytopic factors from the corresponding observations. This article aims to characterize all eligible polytopic sets that are suitable for the PMF framework. In particular, we show that any polytope whose set of vertices have only permutation and/or sign invariances qualifies for PMF framework. Such a rich set of possibilities enables elastic modeling of independent/dependent latent factors with combination of features such as relatively sparse/antisparse subvectors, mixture of signed/nonnegative components with optionally prescribed domains.