Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability

TL;DR

This paper introduces Polytopic Matrix Factorization (PMF), a new data decomposition method modeling data as transformations of latent vectors within a polytope, using determinant maximization for identifiability and flexible feature encoding.

Contribution

It proposes a novel PMF framework with a determinant maximization criterion and a sufficient identifiability condition, allowing flexible latent feature modeling through polytope choices.

Findings

Identifiability condition based on maximum volume inscribed ellipsoid.

PMF framework applies to polytopes with symmetry restrictions.

Flexible latent feature representation including nonnegativity and sparsity.

Abstract

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.